Integrand size = 24, antiderivative size = 845 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {149 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {149 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
1/10*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-3/20*a*c*x*(a^2*c*x^2+c)^(3/2)*arct an(a*x)^2+1/3*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3+1/5*(a^2*c*x^2+c)^(5/2)* arctan(a*x)^3-3/2*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))-6*I*c^3 *polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^( 1/2)-2*c^3*arctan(a*x)^3*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^ (1/2)/(a^2*c*x^2+c)^(1/2)-3*I*c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x ^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+3*I*c^3*arctan(a*x)^2*p olylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/ 2)-149/20*I*c^3*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2 *x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*I*c^3*polylog(4,(1+I*a*x)/(a^2*x^2+1)^ (1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*c^3*arctan(a*x)*polylog(3,- (1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+149/20* c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2 +c)^(1/2)-149/20*c^3*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^ (1/2)/(a^2*c*x^2+c)^(1/2)+6*c^3*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1 )^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+149/20*I*c^3*arctan((1+I*a* x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+ 149/20*I*c^3*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2 +1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/20*a*c^2*x*(a^2*c*x^2+c)^(1/2)+29/20*c^2*a rctan(a*x)*(a^2*c*x^2+c)^(1/2)-29/40*a*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c...
Time = 5.86 (sec) , antiderivative size = 723, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (-120 i \pi ^4+960 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)-150 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)+1392 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+960 \sqrt {1+a^2 x^2} \arctan (a x)^3+640 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^3+32 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^3+240 i \arctan (a x)^4-1440 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+960 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \cos (2 \arctan (a x))-216 \left (1+a^2 x^2\right )^{5/2} \arctan (a x) \cos (2 \arctan (a x))-160 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^3 \cos (2 \arctan (a x))-66 \left (1+a^2 x^2\right )^{5/2} \arctan (a x) \cos (4 \arctan (a x))+960 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-2880 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+2880 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-960 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+2880 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+2880 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-7152 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+7152 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+5760 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-5760 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+7152 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-7152 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )-5760 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-5760 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )-12 \left (1+a^2 x^2\right )^{5/2} \sin (2 \arctan (a x))-480 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \sin (2 \arctan (a x))-6 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^2 \sin (2 \arctan (a x))-6 \left (1+a^2 x^2\right )^{5/2} \sin (4 \arctan (a x))+33 \left (1+a^2 x^2\right )^{5/2} \arctan (a x)^2 \sin (4 \arctan (a x))\right )}{960 \sqrt {1+a^2 x^2}} \]
(c^2*Sqrt[c + a^2*c*x^2]*((-120*I)*Pi^4 + 960*(1 + a^2*x^2)^(3/2)*ArcTan[a *x] - 150*(1 + a^2*x^2)^(5/2)*ArcTan[a*x] + (1392*I)*ArcTan[E^(I*ArcTan[a* x])]*ArcTan[a*x]^2 + 960*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 640*(1 + a^2*x^ 2)^(3/2)*ArcTan[a*x]^3 + 32*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^3 + (240*I)*Ar cTan[a*x]^4 - 1440*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 960*(1 + a^2*x^2)^(3 /2)*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 216*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]*C os[2*ArcTan[a*x]] - 160*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^3*Cos[2*ArcTan[a*x ]] - 66*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]*Cos[4*ArcTan[a*x]] + 960*ArcTan[a* x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 2880*ArcTan[a*x]^2*Log[1 - I*E^(I*Arc Tan[a*x])] + 2880*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 960*ArcTan[ a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (2880*I)*ArcTan[a*x]^2*PolyLog[2, E^(( -I)*ArcTan[a*x])] + (2880*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] - (7152*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (7152*I)*ArcTa n[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 5760*ArcTan[a*x]*PolyLog[3, E^((- I)*ArcTan[a*x])] - 5760*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + 7152* PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 7152*PolyLog[3, I*E^(I*ArcTan[a*x])] - (5760*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (5760*I)*PolyLog[4, -E^(I*Ar cTan[a*x])] - 12*(1 + a^2*x^2)^(5/2)*Sin[2*ArcTan[a*x]] - 480*(1 + a^2*x^2 )^(3/2)*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] - 6*(1 + a^2*x^2)^(5/2)*ArcTan[a* x]^2*Sin[2*ArcTan[a*x]] - 6*(1 + a^2*x^2)^(5/2)*Sin[4*ArcTan[a*x]] + 33...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{x} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3dx+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (a^2 c \int x \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx\right )+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {3 \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )}{5 a}\right )+c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+a^2 c \left (\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}}{a}\right )\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (\frac {\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}{5 a^2 c}-\frac {3 \left (\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{6 a}+\frac {1}{6} c \left (\frac {1}{2} \sqrt {a^2 c x^2+c} x+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}\right )+\frac {3}{4} c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right )\right )}{5 a}\right ) a^2+c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}}{a}\right ) a^2+c \left (c \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx a^2+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (\frac {\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}{5 a^2 c}-\frac {3 \left (\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{6 a}+\frac {1}{6} c \left (\frac {1}{2} \sqrt {a^2 c x^2+c} x+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}\right )+\frac {3}{4} c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right )\right )}{5 a}\right ) a^2+c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}}{a}\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right ) a^2+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (\frac {\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}{5 a^2 c}-\frac {3 \left (\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{6 a}+\frac {1}{6} c \left (\frac {1}{2} \sqrt {a^2 c x^2+c} x+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}\right )+\frac {3}{4} c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right )\right )}{5 a}\right ) a^2+c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}}{a}\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right ) a^2+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (\frac {\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}{5 a^2 c}-\frac {3 \left (\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{6 a}+\frac {1}{6} c \left (\frac {1}{2} \sqrt {a^2 c x^2+c} x+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}\right )+\frac {3}{4} c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right )\right )}{5 a}\right ) a^2+c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}}{a}\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
3.5.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt Q[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan[c*x])^ p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & & IGtQ[p, 0] && !GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Time = 7.49 (sec) , antiderivative size = 562, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 a^{4} \arctan \left (a x \right )^{3} x^{4}-18 a^{3} \arctan \left (a x \right )^{2} x^{3}+88 \arctan \left (a x \right )^{3} x^{2} a^{2}+12 a^{2} \arctan \left (a x \right ) x^{2}-105 a \arctan \left (a x \right )^{2} x +184 \arctan \left (a x \right )^{3}-6 a x +186 \arctan \left (a x \right )\right )}{120}-\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (40 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-40 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+120 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-149 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+149 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+298 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-298 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+240 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-240 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+240 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-240 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-298 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+298 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40 \sqrt {a^{2} x^{2}+1}}\) | \(562\) |
1/120*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(24*a^4*arctan(a*x)^3*x^4-18*a^3*arcta n(a*x)^2*x^3+88*arctan(a*x)^3*x^2*a^2+12*a^2*arctan(a*x)*x^2-105*a*arctan( a*x)^2*x+184*arctan(a*x)^3-6*a*x+186*arctan(a*x))-1/40*c^2*(c*(a*x-I)*(I+a *x))^(1/2)*(40*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-40*arctan(a *x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-120*I*arctan(a*x)^2*polylog(2,-(1+ I*a*x)/(a^2*x^2+1)^(1/2))+120*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2 +1)^(1/2))-149*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+149*arcta n(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+298*I*arctan(a*x)*polylog(2,- I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-298*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^ 2*x^2+1)^(1/2))+240*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-24 0*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+240*I*polylog(4,-(1+I *a*x)/(a^2*x^2+1)^(1/2))-240*I*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-120* I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))-298*polylog(3,-I*(1+I*a*x)/(a^2*x^2+ 1)^(1/2))+298*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \]
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x} \,d x \]