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1 / | | 2 | (cos(a*x)*cos(b*x)) dx | / 0
$$\int\limits_{0}^{1} \left(\cos{\left(a x \right)} \cos{\left(b x \right)}\right)^{2}\, dx$$
Integral((cos(a*x)*cos(b*x))^2, (x, 0, 1))
The answer
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/ 1 for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0)) | | 2 2 | cos (b) sin (b) cos(b)*sin(b) | ------- + ------- + ------------- for Or(And(a = 0, a = -b), And(a = 0, a = b), And(a = 0, a = -b, a = b), a = 0) | 2 2 2*b | | 4 4 2 2 3 3 | 3*cos (b) 3*sin (b) 3*cos (b)*sin (b) 3*sin (b)*cos(b) 5*cos (b)*sin(b) | --------- + --------- + ----------------- + ---------------- + ---------------- for Or(And(a = -b, a = b), And(a = b, b = 0), And(a = -b, b = 0), And(a = -b, a = b, b = 0), a = -b, a = b) | 8 8 4 8*b 8*b < | 2 2 | cos (a) sin (a) cos(a)*sin(a) | ------- + ------- + ------------- for b = 0 | 2 2 2*a | | 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 2 2 2 |b*a *cos (a)*cos (b) b*a *cos (a)*sin (b) b*a *cos (b)*sin (a) b*a *sin (a)*sin (b) a *cos (a)*cos(b)*sin(b) a *sin (a)*cos(b)*sin(b) a*b *cos (a)*cos (b) a*b *cos (a)*sin (b) a*b *cos (b)*sin (a) a*b *sin (a)*sin (b) b *cos (b)*cos(a)*sin(a) b *sin (b)*cos(a)*sin(a) 2*a*b *cos (a)*cos(b)*sin(b) 2*b*a *cos (b)*cos(a)*sin(a) |-------------------- + -------------------- + -------------------- + -------------------- + ------------------------ + ------------------------ - -------------------- - -------------------- - -------------------- - -------------------- - ------------------------ - ------------------------ - ---------------------------- + ---------------------------- otherwise | 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 | - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a \
$$\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = 0 \wedge a = b\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee a = 0 \\\frac{3 \sin^{4}{\left(b \right)}}{8} + \frac{3 \sin^{2}{\left(b \right)} \cos^{2}{\left(b \right)}}{4} + \frac{3 \cos^{4}{\left(b \right)}}{8} + \frac{3 \sin^{3}{\left(b \right)} \cos{\left(b \right)}}{8 b} + \frac{5 \sin{\left(b \right)} \cos^{3}{\left(b \right)}}{8 b} & \text{for}\: \left(a = - b \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \vee a = b \\\frac{\sin^{2}{\left(a \right)}}{2} + \frac{\cos^{2}{\left(a \right)}}{2} + \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b \sin^{2}{\left(a \right)} \sin^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b \sin^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b \sin^{2}{\left(b \right)} \cos^{2}{\left(a \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b \cos^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin^{2}{\left(a \right)} \sin{\left(b \right)} \cos{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin{\left(b \right)} \cos^{2}{\left(a \right)} \cos{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{2 a^{2} b \sin{\left(a \right)} \cos{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \sin^{2}{\left(a \right)} \sin^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \sin^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \sin^{2}{\left(b \right)} \cos^{2}{\left(a \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \cos^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{2 a b^{2} \sin{\left(b \right)} \cos^{2}{\left(a \right)} \cos{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a \right)} \sin^{2}{\left(b \right)} \cos{\left(a \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a \right)} \cos{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} & \text{otherwise} \end{cases}$$
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/ 1 for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0)) | | 2 2 | cos (b) sin (b) cos(b)*sin(b) | ------- + ------- + ------------- for Or(And(a = 0, a = -b), And(a = 0, a = b), And(a = 0, a = -b, a = b), a = 0) | 2 2 2*b | | 4 4 2 2 3 3 | 3*cos (b) 3*sin (b) 3*cos (b)*sin (b) 3*sin (b)*cos(b) 5*cos (b)*sin(b) | --------- + --------- + ----------------- + ---------------- + ---------------- for Or(And(a = -b, a = b), And(a = b, b = 0), And(a = -b, b = 0), And(a = -b, a = b, b = 0), a = -b, a = b) | 8 8 4 8*b 8*b < | 2 2 | cos (a) sin (a) cos(a)*sin(a) | ------- + ------- + ------------- for b = 0 | 2 2 2*a | | 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 2 2 2 |b*a *cos (a)*cos (b) b*a *cos (a)*sin (b) b*a *cos (b)*sin (a) b*a *sin (a)*sin (b) a *cos (a)*cos(b)*sin(b) a *sin (a)*cos(b)*sin(b) a*b *cos (a)*cos (b) a*b *cos (a)*sin (b) a*b *cos (b)*sin (a) a*b *sin (a)*sin (b) b *cos (b)*cos(a)*sin(a) b *sin (b)*cos(a)*sin(a) 2*a*b *cos (a)*cos(b)*sin(b) 2*b*a *cos (b)*cos(a)*sin(a) |-------------------- + -------------------- + -------------------- + -------------------- + ------------------------ + ------------------------ - -------------------- - -------------------- - -------------------- - -------------------- - ------------------------ - ------------------------ - ---------------------------- + ---------------------------- otherwise | 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 | - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a \
$$\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = 0 \wedge a = b\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee a = 0 \\\frac{3 \sin^{4}{\left(b \right)}}{8} + \frac{3 \sin^{2}{\left(b \right)} \cos^{2}{\left(b \right)}}{4} + \frac{3 \cos^{4}{\left(b \right)}}{8} + \frac{3 \sin^{3}{\left(b \right)} \cos{\left(b \right)}}{8 b} + \frac{5 \sin{\left(b \right)} \cos^{3}{\left(b \right)}}{8 b} & \text{for}\: \left(a = - b \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \vee a = b \\\frac{\sin^{2}{\left(a \right)}}{2} + \frac{\cos^{2}{\left(a \right)}}{2} + \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b \sin^{2}{\left(a \right)} \sin^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b \sin^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b \sin^{2}{\left(b \right)} \cos^{2}{\left(a \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b \cos^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin^{2}{\left(a \right)} \sin{\left(b \right)} \cos{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin{\left(b \right)} \cos^{2}{\left(a \right)} \cos{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{2 a^{2} b \sin{\left(a \right)} \cos{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \sin^{2}{\left(a \right)} \sin^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \sin^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \sin^{2}{\left(b \right)} \cos^{2}{\left(a \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} \cos^{2}{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{2 a b^{2} \sin{\left(b \right)} \cos^{2}{\left(a \right)} \cos{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a \right)} \sin^{2}{\left(b \right)} \cos{\left(a \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a \right)} \cos{\left(a \right)} \cos^{2}{\left(b \right)}}{4 a^{3} b - 4 a b^{3}} & \text{otherwise} \end{cases}$$
Piecewise((1, ((a = 0)∧(b = 0))∨((a = 0)∧(a = b)∧(b = 0))∨((a = 0)∧(b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(b = 0)∧(a = -b))), (cos(b)^2/2 + sin(b)^2/2 + cos(b)*sin(b)/(2*b), (a = 0)∨((a = 0)∧(a = b))∨((a = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(a = -b))), (3*cos(b)^4/8 + 3*sin(b)^4/8 + 3*cos(b)^2*sin(b)^2/4 + 3*sin(b)^3*cos(b)/(8*b) + 5*cos(b)^3*sin(b)/(8*b), (a = b)∨(a = -b)∨((a = b)∧(b = 0))∨((a = b)∧(a = -b))∨((b = 0)∧(a = -b))∨((a = b)∧(b = 0)∧(a = -b))), (cos(a)^2/2 + sin(a)^2/2 + cos(a)*sin(a)/(2*a), b = 0), (b*a^3*cos(a)^2*cos(b)^2/(-4*a*b^3 + 4*b*a^3) + b*a^3*cos(a)^2*sin(b)^2/(-4*a*b^3 + 4*b*a^3) + b*a^3*cos(b)^2*sin(a)^2/(-4*a*b^3 + 4*b*a^3) + b*a^3*sin(a)^2*sin(b)^2/(-4*a*b^3 + 4*b*a^3) + a^3*cos(a)^2*cos(b)*sin(b)/(-4*a*b^3 + 4*b*a^3) + a^3*sin(a)^2*cos(b)*sin(b)/(-4*a*b^3 + 4*b*a^3) - a*b^3*cos(a)^2*cos(b)^2/(-4*a*b^3 + 4*b*a^3) - a*b^3*cos(a)^2*sin(b)^2/(-4*a*b^3 + 4*b*a^3) - a*b^3*cos(b)^2*sin(a)^2/(-4*a*b^3 + 4*b*a^3) - a*b^3*sin(a)^2*sin(b)^2/(-4*a*b^3 + 4*b*a^3) - b^3*cos(b)^2*cos(a)*sin(a)/(-4*a*b^3 + 4*b*a^3) - b^3*sin(b)^2*cos(a)*sin(a)/(-4*a*b^3 + 4*b*a^3) - 2*a*b^2*cos(a)^2*cos(b)*sin(b)/(-4*a*b^3 + 4*b*a^3) + 2*b*a^2*cos(b)^2*cos(a)*sin(a)/(-4*a*b^3 + 4*b*a^3), True))
Use the examples entering the upper and lower limits of integration.