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}} \] Output:
-1/20*a*c^2*x*(a^2*c*x^2+c)^(1/2)+29/20*c^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x )+1/10*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-29/40*a*c^2*x*(a^2*c*x^2+c)^(1/2) *arctan(a*x)^2-3/20*a*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+3*I*c^3*(a^2*x ^2+1)^(1/2)*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x ^2+c)^(1/2)+c^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3+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)^3-2*c^3*(a^2*x^2+1)^(1 /2)*arctan(a*x)^3*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2) -3/2*c^(5/2)*arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))+149/20*I*c^3*(a^2*x^ 2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2/(a^2*c*x^2+c) ^(1/2)+149/20*I*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a ^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-149/20*I*c^3*(a^2*x^2+1)^(1/2)*arctan (a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-6*I*c^ 3*(a^2*x^2+1)^(1/2)*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^ (1/2)-6*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1) ^(1/2))/(a^2*c*x^2+c)^(1/2)+149/20*c^3*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I *a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-149/20*c^3*(a^2*x^2+1)^(1/2)* polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*c^3*(a^2*x^ 2+1)^(1/2)*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c )^(1/2)-3*I*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x ^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*I*c^3*(a^2*x^2+1)^(1/2)*polylog(4,(1...
Time = 4.81 (sec) , antiderivative size = 723, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx =\text {Too large to display} \] Input:
Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x,x]
Output:
(c^2*Sqrt[c + a^2*c*x^2]*((-120*I)*Pi^4 - 1440*ArcCoth[(a*x)/Sqrt[1 + a^2* x^2]] + 960*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] - 150*(1 + a^2*x^2)^(5/2)*ArcT an[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)*ArcTan[a*x]^4 + 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 )\) |
Input:
Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x,x]
Output:
$Aborted
Time = 8.79 (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 \arctan \left (a x \right )^{3} a^{4} x^{4}-18 a^{3} \arctan \left (a x \right )^{2} x^{3}+88 \arctan \left (a x \right )^{3} a^{2} x^{2}+12 x^{2} a^{2} \arctan \left (a x \right )-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 (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 )+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 )+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 )-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 )-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 \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 )+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}}\) | \(562\) |
Input:
int((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,x,method=_RETURNVERBOSE)
Output:
1/120*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(24*arctan(a*x)^3*a^4*x^4-18*a^3*arcta n(a*x)^2*x^3+88*arctan(a*x)^3*a^2*x^2+12*x^2*a^2*arctan(a*x)-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)*(a*x +I))^(1/2)*(-40*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+120*I*arct an(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+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))+149*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-298*I*ar ctan(a*x)*polylog(2,-I*(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))+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))-2 40*I*polylog(4,-(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))+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)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,x, algorithm="fricas")
Output:
integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a* x)^3/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx \] Input:
integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)**3/x,x)
Output:
Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}{x} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3/x, x)
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,x, algorithm="giac")
Output:
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. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x} \,d x \] Input:
int((atan(a*x)^3*(c + a^2*c*x^2)^(5/2))/x,x)
Output:
int((atan(a*x)^3*(c + a^2*c*x^2)^(5/2))/x, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x} \, dx=\sqrt {c}\, c^{2} \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}}{x}d x +\left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{3}d x \right ) a^{4}+2 \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x d x \right ) a^{2}\right ) \] Input:
int((a^2*c*x^2+c)^(5/2)*atan(a*x)^3/x,x)
Output:
sqrt(c)*c**2*(int((sqrt(a**2*x**2 + 1)*atan(a*x)**3)/x,x) + int(sqrt(a**2* x**2 + 1)*atan(a*x)**3*x**3,x)*a**4 + 2*int(sqrt(a**2*x**2 + 1)*atan(a*x)* *3*x,x)*a**2)