Integral of arctg*sqrt(4x-1) dx

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                                   |     __________       |   __________                           
                                   | x*\/ -1 + 4*x        | \/ -1 + 4*x                            
                                2* | -------------- dx    | ------------ dx                        
                                   |          2           |         2                              
  /                                |     1 + x            |    1 + x                               
 |                                 |                      |                             3/2        
 |           _________            /                      /                    (-1 + 4*x)   *atan(x)
 | atan(x)*\/ 4*x - 1  dx = C - ---------------------- + ------------------ + ---------------------
 |                                        3                      6                      6          
/                                                                                                  

$$-{{2448\,\int {{{\sqrt{4\,x-1}\,\left(\left(4\,x-1\right)\,\log \left(x^2+1\right)+\left(2\,x-8\,x^2\right)\,\arctan x\right)}\over{ 48\,x^2+48}}}{\;dx}-4896\,\int {{{\sqrt{4\,x-1}\,\left(x\,\log \left(x^2+1\right)+2\,\arctan x\right)}\over{24\,x^2+24}}}{\;dx}+ 2448\,\int {{{\sqrt{4\,x-1}\,\left(\log \left(x^2+1\right)-2\,x\, \arctan x\right)}\over{48\,x^2+48}}}{\;dx}+\sqrt{\sqrt{17}-1}\, \left(30\,\sqrt{34}\,{\rm atan2}\left({{\sqrt{\sqrt{17}+1}\,\left( \sqrt{\sqrt{17}-1}\,\left(2^{{{5}\over{2}}}\,x-\sqrt{2}\right)+ \sqrt{4\,x-1}\,\left(4\,x+\sqrt{17}-1\right)\right)}\over{2^{{{7 }\over{2}}}\,x^2+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(8\,x+2\, \sqrt{17}-2\right)+\left(2^{{{7}\over{2}}}\,\sqrt{17}-2^{{{7}\over{2 }}}\right)\,x-2^{{{3}\over{2}}}\,\sqrt{17}+5\,2^{{{3}\over{2}}}}} , {{2^{{{7}\over{2}}}\,x^2+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(4 \,x-\sqrt{17}-1\right)-2^{{{5}\over{2}}}\,x-2^{{{7}\over{2}}}}\over{ 2^{{{7}\over{2}}}\,x^2+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(8\,x +2\,\sqrt{17}-2\right)+\left(2^{{{7}\over{2}}}\,\sqrt{17}-2^{{{7 }\over{2}}}\right)\,x-2^{{{3}\over{2}}}\,\sqrt{17}+5\,2^{{{3}\over{2 }}}}}\right)-8\,\sqrt{34}\,\log \left({{8\,x^{14}+\sqrt{\sqrt{17}-1} \,\sqrt{4\,x-1}\,\left(3\,2^{{{7}\over{2}}}\,x^{13}+\left(143\,2^{{{ 3}\over{2}}}\,\sqrt{17}-143\,2^{{{3}\over{2}}}\right)\,x^{12}+\left( 8801\,2^{{{3}\over{2}}}-1001\,2^{{{3}\over{2}}}\,\sqrt{17}\right)\,x ^{11}+\left(29887\,\sqrt{2}\,\sqrt{17}-81367\,\sqrt{2}\right)\,x^{10 }+\left(243529\,2^{{{3}\over{2}}}-47333\,2^{{{3}\over{2}}}\,\sqrt{17 }\right)\,x^9+\left(293553\,\sqrt{2}\,\sqrt{17}-1075945\,\sqrt{2} \right)\,x^8+\left(512635\,2^{{{5}\over{2}}}-116259\,2^{{{5}\over{2 }}}\,\sqrt{17}\right)\,x^7+\left(308389\,2^{{{3}\over{2}}}\,\sqrt{17 }-1222125\,2^{{{3}\over{2}}}\right)\,x^6+\left(302167\,2^{{{7}\over{ 2}}}-35987\,2^{{{9}\over{2}}}\,\sqrt{17}\right)\,x^5+\left(211291\,2 ^{{{3}\over{2}}}\,\sqrt{17}-868747\,2^{{{3}\over{2}}}\right)\,x^4+ \left(111231\,2^{{{7}\over{2}}}-27051\,2^{{{7}\over{2}}}\,\sqrt{17} \right)\,x^3+\left(17689\,2^{{{5}\over{2}}}\,\sqrt{17}-73057\,2^{{{5 }\over{2}}}\right)\,x^2+\left(13393\,2^{{{5}\over{2}}}-191\,2^{{{5 }\over{2}}}\,17^{{{3}\over{2}}}\right)\,x+125\,2^{{{7}\over{2}}}\, \sqrt{17}-515\,2^{{{7}\over{2}}}\right)+\left(288\,\sqrt{17}-288 \right)\,x^{13}+\left(30368-3432\,\sqrt{17}\right)\,x^{12}+\left( 75088\,\sqrt{17}-203216\right)\,x^{11}+\left(1648218-320034\,\sqrt{ 17}\right)\,x^{10}+\left(1274416\,\sqrt{17}-4697264\right)\,x^9+ \left(10983258-2515058\,\sqrt{17}\right)\,x^8+\left(231648\,17^{{{3 }\over{2}}}-15729120\right)\,x^7+\left(18074232-4316384\,\sqrt{17} \right)\,x^6+\left(3764256\,\sqrt{17}-15472160\right)\,x^5+\left( 9946992-2418120\,\sqrt{17}\right)\,x^4+\left(63168\,17^{{{3}\over{2 }}}-4435136\right)\,x^3+\left(1251352-303392\,\sqrt{17}\right)\,x^2+ \left(48000\,\sqrt{17}-197760\right)\,x-3200\,\sqrt{17}+13192}\over{ 8\,x^{14}+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(7\,2^{{{5}\over{2 }}}\,x^{13}+\left(455\,\sqrt{2}\,\sqrt{17}-455\,\sqrt{2}\right)\,x^{ 12}+\left(4823\,2^{{{7}\over{2}}}-273\,2^{{{9}\over{2}}}\,\sqrt{17} \right)\,x^{11}+\left(45617\,2^{{{3}\over{2}}}\,\sqrt{17}-123409\,2 ^{{{3}\over{2}}}\right)\,x^{10}+\left(32461\,2^{{{13}\over{2}}}- 100815\,2^{{{5}\over{2}}}\,\sqrt{17}\right)\,x^9+\left(1724983\, \sqrt{2}\,\sqrt{17}-6388967\,\sqrt{2}\right)\,x^8+\left(989841\,2^{ {{9}\over{2}}}-114855\,2^{{{11}\over{2}}}\,\sqrt{17}\right)\,x^7+ \left(1424899\,2^{{{5}\over{2}}}\,\sqrt{17}-5796483\,2^{{{5}\over{2 }}}\right)\,x^6+\left(11710237\,2^{{{3}\over{2}}}-2835987\,2^{{{3 }\over{2}}}\,\sqrt{17}\right)\,x^5+\left(118097\,2^{{{11}\over{2}}} \,\sqrt{17}-1950207\,2^{{{7}\over{2}}}\right)\,x^4+\left(1691527\,2 ^{{{5}\over{2}}}-410549\,2^{{{5}\over{2}}}\,\sqrt{17}\right)\,x^3+ \left(110769\,2^{{{5}\over{2}}}\,\sqrt{17}-456661\,2^{{{5}\over{2}}} \right)\,x^2+\left(34575\,2^{{{7}\over{2}}}-16767\,2^{{{5}\over{2}}} \,\sqrt{17}\right)\,x+4329\,\sqrt{2}\,\sqrt{17}-17849\,\sqrt{2} \right)+\left(392\,\sqrt{17}-392\right)\,x^{13}+\left(56602-6370\, \sqrt{17}\right)\,x^{12}+\left(193648\,\sqrt{17}-519792\right)\,x^{ 11}+\left(5921916-1141140\,\sqrt{17}\right)\,x^{10}+\left(6350344\, \sqrt{17}-23391368\right)\,x^9+\left(73811738-17070690\,\sqrt{17} \right)\,x^8+\left(33390496\,\sqrt{17}-135600032\right)\,x^7+\left( 173087908-41892844\,\sqrt{17}\right)\,x^6+\left(35761152\,\sqrt{17}- 147622720\right)\,x^5+\left(84483084-20505716\,\sqrt{17}\right)\,x^4 +\left(7717472\,\sqrt{17}-31817184\right)\,x^3+\left(7504616-1819664 \,\sqrt{17}\right)\,x^2+\left(242424\,\sqrt{17}-999544\right)\,x- 13858\,\sqrt{17}+57130}}\right)\right)+\sqrt{\sqrt{17}+1}\,\left(16 \,\sqrt{34}\,{\rm atan2}\left({{\sqrt{\sqrt{17}+1}\,\left(\sqrt{ \sqrt{17}-1}\,\left(2^{{{5}\over{2}}}\,x-\sqrt{2}\right)+\sqrt{4\,x- 1}\,\left(4\,x+\sqrt{17}-1\right)\right)}\over{2^{{{7}\over{2}}}\,x^ 2+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(8\,x+2\,\sqrt{17}-2 \right)+\left(2^{{{7}\over{2}}}\,\sqrt{17}-2^{{{7}\over{2}}}\right) \,x-2^{{{3}\over{2}}}\,\sqrt{17}+5\,2^{{{3}\over{2}}}}} , {{2^{{{7 }\over{2}}}\,x^2+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(4\,x- \sqrt{17}-1\right)-2^{{{5}\over{2}}}\,x-2^{{{7}\over{2}}}}\over{2^{ {{7}\over{2}}}\,x^2+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\,\left(8\,x+2 \,\sqrt{17}-2\right)+\left(2^{{{7}\over{2}}}\,\sqrt{17}-2^{{{7 }\over{2}}}\right)\,x-2^{{{3}\over{2}}}\,\sqrt{17}+5\,2^{{{3}\over{2 }}}}}\right)+15\,\sqrt{34}\,\log \left({{8\,x^{14}+\sqrt{\sqrt{17}-1 }\,\sqrt{4\,x-1}\,\left(3\,2^{{{7}\over{2}}}\,x^{13}+\left(143\,2^{ {{3}\over{2}}}\,\sqrt{17}-143\,2^{{{3}\over{2}}}\right)\,x^{12}+ \left(8801\,2^{{{3}\over{2}}}-1001\,2^{{{3}\over{2}}}\,\sqrt{17} \right)\,x^{11}+\left(29887\,\sqrt{2}\,\sqrt{17}-81367\,\sqrt{2} \right)\,x^{10}+\left(243529\,2^{{{3}\over{2}}}-47333\,2^{{{3}\over{ 2}}}\,\sqrt{17}\right)\,x^9+\left(293553\,\sqrt{2}\,\sqrt{17}- 1075945\,\sqrt{2}\right)\,x^8+\left(512635\,2^{{{5}\over{2}}}-116259 \,2^{{{5}\over{2}}}\,\sqrt{17}\right)\,x^7+\left(308389\,2^{{{3 }\over{2}}}\,\sqrt{17}-1222125\,2^{{{3}\over{2}}}\right)\,x^6+\left( 302167\,2^{{{7}\over{2}}}-35987\,2^{{{9}\over{2}}}\,\sqrt{17}\right) \,x^5+\left(211291\,2^{{{3}\over{2}}}\,\sqrt{17}-868747\,2^{{{3 }\over{2}}}\right)\,x^4+\left(111231\,2^{{{7}\over{2}}}-27051\,2^{{{ 7}\over{2}}}\,\sqrt{17}\right)\,x^3+\left(17689\,2^{{{5}\over{2}}}\, \sqrt{17}-73057\,2^{{{5}\over{2}}}\right)\,x^2+\left(13393\,2^{{{5 }\over{2}}}-191\,2^{{{5}\over{2}}}\,17^{{{3}\over{2}}}\right)\,x+125 \,2^{{{7}\over{2}}}\,\sqrt{17}-515\,2^{{{7}\over{2}}}\right)+\left( 288\,\sqrt{17}-288\right)\,x^{13}+\left(30368-3432\,\sqrt{17}\right) \,x^{12}+\left(75088\,\sqrt{17}-203216\right)\,x^{11}+\left(1648218- 320034\,\sqrt{17}\right)\,x^{10}+\left(1274416\,\sqrt{17}-4697264 \right)\,x^9+\left(10983258-2515058\,\sqrt{17}\right)\,x^8+\left( 231648\,17^{{{3}\over{2}}}-15729120\right)\,x^7+\left(18074232- 4316384\,\sqrt{17}\right)\,x^6+\left(3764256\,\sqrt{17}-15472160 \right)\,x^5+\left(9946992-2418120\,\sqrt{17}\right)\,x^4+\left( 63168\,17^{{{3}\over{2}}}-4435136\right)\,x^3+\left(1251352-303392\, \sqrt{17}\right)\,x^2+\left(48000\,\sqrt{17}-197760\right)\,x-3200\, \sqrt{17}+13192}\over{8\,x^{14}+\sqrt{\sqrt{17}-1}\,\sqrt{4\,x-1}\, \left(7\,2^{{{5}\over{2}}}\,x^{13}+\left(455\,\sqrt{2}\,\sqrt{17}- 455\,\sqrt{2}\right)\,x^{12}+\left(4823\,2^{{{7}\over{2}}}-273\,2^{ {{9}\over{2}}}\,\sqrt{17}\right)\,x^{11}+\left(45617\,2^{{{3}\over{2 }}}\,\sqrt{17}-123409\,2^{{{3}\over{2}}}\right)\,x^{10}+\left(32461 \,2^{{{13}\over{2}}}-100815\,2^{{{5}\over{2}}}\,\sqrt{17}\right)\,x^ 9+\left(1724983\,\sqrt{2}\,\sqrt{17}-6388967\,\sqrt{2}\right)\,x^8+ \left(989841\,2^{{{9}\over{2}}}-114855\,2^{{{11}\over{2}}}\,\sqrt{17 }\right)\,x^7+\left(1424899\,2^{{{5}\over{2}}}\,\sqrt{17}-5796483\,2 ^{{{5}\over{2}}}\right)\,x^6+\left(11710237\,2^{{{3}\over{2}}}- 2835987\,2^{{{3}\over{2}}}\,\sqrt{17}\right)\,x^5+\left(118097\,2^{ {{11}\over{2}}}\,\sqrt{17}-1950207\,2^{{{7}\over{2}}}\right)\,x^4+ \left(1691527\,2^{{{5}\over{2}}}-410549\,2^{{{5}\over{2}}}\,\sqrt{17 }\right)\,x^3+\left(110769\,2^{{{5}\over{2}}}\,\sqrt{17}-456661\,2^{ {{5}\over{2}}}\right)\,x^2+\left(34575\,2^{{{7}\over{2}}}-16767\,2^{ {{5}\over{2}}}\,\sqrt{17}\right)\,x+4329\,\sqrt{2}\,\sqrt{17}-17849 \,\sqrt{2}\right)+\left(392\,\sqrt{17}-392\right)\,x^{13}+\left( 56602-6370\,\sqrt{17}\right)\,x^{12}+\left(193648\,\sqrt{17}-519792 \right)\,x^{11}+\left(5921916-1141140\,\sqrt{17}\right)\,x^{10}+ \left(6350344\,\sqrt{17}-23391368\right)\,x^9+\left(73811738- 17070690\,\sqrt{17}\right)\,x^8+\left(33390496\,\sqrt{17}-135600032 \right)\,x^7+\left(173087908-41892844\,\sqrt{17}\right)\,x^6+\left( 35761152\,\sqrt{17}-147622720\right)\,x^5+\left(84483084-20505716\, \sqrt{17}\right)\,x^4+\left(7717472\,\sqrt{17}-31817184\right)\,x^3+ \left(7504616-1819664\,\sqrt{17}\right)\,x^2+\left(242424\,\sqrt{17} -999544\right)\,x-13858\,\sqrt{17}+57130}}\right)\right)+\sqrt{4\,x- 1}\,\left(\left(68-272\,x\right)\,\arctan x+544\right)}\over{816}}$$

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