Integral of tan^3x×sec^3x dx

1. Rewrite the integrand:

   $$\tan^{3}{\left(x \right)} \sec^{3}{\left(x \right)} = \left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec^{3}{\left(x \right)}$$

2. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sec{\left(x \right)}$.

      Then let $du = \tan{\left(x \right)} \sec{\left(x \right)} dx$ and substitute $du$:

      $$\int \left(u^{4} - u^{2}\right)\, du$$

      1. Integrate term-by-term:

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int u^{4}\, du = \frac{u^{5}}{5}$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         The result is: $\frac{u^{5}}{5} - \frac{u^{3}}{3}$

      Now substitute $u$ back in:

      $$\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec^{3}{\left(x \right)} = \tan{\left(x \right)} \sec^{5}{\left(x \right)} - \tan{\left(x \right)} \sec^{3}{\left(x \right)}$$

   2. Integrate term-by-term:

      1. Let $u = \sec^{5}{\left(x \right)}$.

         Then let $du = 5 \tan{\left(x \right)} \sec^{5}{\left(x \right)} dx$ and substitute $\frac{du}{5}$:

         $$\int \frac{1}{25}\, du$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}$$

            1. The integral of a constant is the constant times the variable of integration:

               $$\int 1\, du = u$$

            So, the result is: $\frac{u}{5}$

         Now substitute $u$ back in:

         $$\frac{\sec^{5}{\left(x \right)}}{5}$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \tan{\left(x \right)} \sec^{3}{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)} \sec^{3}{\left(x \right)}\, dx$$

         1. Let $u = \sec^{3}{\left(x \right)}$.

            Then let $du = 3 \tan{\left(x \right)} \sec^{3}{\left(x \right)} dx$ and substitute $\frac{du}{3}$:

            $$\int \frac{1}{9}\, du$$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \frac{1}{3}\, du = \frac{\int 1\, du}{3}$$

               1. The integral of a constant is the constant times the variable of integration:

                  $$\int 1\, du = u$$

               So, the result is: $\frac{u}{3}$

            Now substitute $u$ back in:

            $$\frac{\sec^{3}{\left(x \right)}}{3}$$

         So, the result is: $- \frac{\sec^{3}{\left(x \right)}}{3}$

      The result is: $\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}$

   Method #3

   1. Rewrite the integrand:

      $$\left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec^{3}{\left(x \right)} = \tan{\left(x \right)} \sec^{5}{\left(x \right)} - \tan{\left(x \right)} \sec^{3}{\left(x \right)}$$

   2. Integrate term-by-term:

      1. Let $u = \sec^{5}{\left(x \right)}$.

         Then let $du = 5 \tan{\left(x \right)} \sec^{5}{\left(x \right)} dx$ and substitute $\frac{du}{5}$:

         $$\int \frac{1}{25}\, du$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}$$

            1. The integral of a constant is the constant times the variable of integration:

               $$\int 1\, du = u$$

            So, the result is: $\frac{u}{5}$

         Now substitute $u$ back in:

         $$\frac{\sec^{5}{\left(x \right)}}{5}$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \tan{\left(x \right)} \sec^{3}{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)} \sec^{3}{\left(x \right)}\, dx$$

         1. Let $u = \sec^{3}{\left(x \right)}$.

            Then let $du = 3 \tan{\left(x \right)} \sec^{3}{\left(x \right)} dx$ and substitute $\frac{du}{3}$:

            $$\int \frac{1}{9}\, du$$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \frac{1}{3}\, du = \frac{\int 1\, du}{3}$$

               1. The integral of a constant is the constant times the variable of integration:

                  $$\int 1\, du = u$$

               So, the result is: $\frac{u}{3}$

            Now substitute $u$ back in:

            $$\frac{\sec^{3}{\left(x \right)}}{3}$$

         So, the result is: $- \frac{\sec^{3}{\left(x \right)}}{3}$

      The result is: $\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}$

3. Add the constant of integration:

   $$\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}+ \mathrm{constant}$$