Optimal. Leaf size=370 \[ -\frac {1}{4} a c^2 x+\frac {1}{4} c^2 \text {ArcTan}(a x)+\frac {1}{4} a^2 c^2 x^2 \text {ArcTan}(a x)-2 i c^2 \text {ArcTan}(a x)^2-\frac {9}{4} a c^2 x \text {ArcTan}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \text {ArcTan}(a x)^2+\frac {3}{4} c^2 \text {ArcTan}(a x)^3+a^2 c^2 x^2 \text {ArcTan}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \text {ArcTan}(a x)^3+2 c^2 \text {ArcTan}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )-2 i c^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \text {ArcTan}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \text {ArcTan}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \text {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
[Out]
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Rubi [A]
time = 0.70, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps
used = 36, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5068,
4942, 5108, 5004, 5114, 5118, 6745, 4946, 5036, 4930, 5040, 4964, 2449, 2352, 327, 209}
\begin {gather*} \frac {1}{4} a^4 c^2 x^4 \text {ArcTan}(a x)^3-\frac {1}{4} a^3 c^2 x^3 \text {ArcTan}(a x)^2+a^2 c^2 x^2 \text {ArcTan}(a x)^3+\frac {1}{4} a^2 c^2 x^2 \text {ArcTan}(a x)-\frac {3}{2} i c^2 \text {ArcTan}(a x)^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+\frac {3}{2} i c^2 \text {ArcTan}(a x)^2 \text {Li}_2\left (\frac {2}{i a x+1}-1\right )-\frac {3}{2} c^2 \text {ArcTan}(a x) \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+\frac {3}{2} c^2 \text {ArcTan}(a x) \text {Li}_3\left (\frac {2}{i a x+1}-1\right )-\frac {9}{4} a c^2 x \text {ArcTan}(a x)^2+\frac {3}{4} c^2 \text {ArcTan}(a x)^3-2 i c^2 \text {ArcTan}(a x)^2+\frac {1}{4} c^2 \text {ArcTan}(a x)-4 c^2 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )+2 c^2 \text {ArcTan}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-2 i c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c^2 \text {Li}_4\left (1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c^2 \text {Li}_4\left (\frac {2}{i a x+1}-1\right )-\frac {1}{4} a c^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4942
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rule 5068
Rule 5108
Rule 5114
Rule 5118
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3}{x} \, dx &=\int \left (\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^4 c^2 x^3 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int \frac {\tan ^{-1}(a x)^3}{x} \, dx+\left (2 a^2 c^2\right ) \int x \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^3 \tan ^{-1}(a x)^3 \, dx\\ &=a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\left (6 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\left (3 a c^2\right ) \int \tan ^{-1}(a x)^2 \, dx+\left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3 c^2\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx+\frac {1}{4} \left (3 a^3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-3 a c^2 x \tan ^{-1}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+c^2 \tan ^{-1}(a x)^3+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (3 i a c^2\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a c^2\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{4} \left (3 a c^2\right ) \int \tan ^{-1}(a x)^2 \, dx-\frac {1}{4} \left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (6 a^2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^4 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-3 i c^2 \tan ^{-1}(a x)^2-\frac {9}{4} a c^2 x \tan ^{-1}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} c^2 \tan ^{-1}(a x)^3+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^2\right ) \int \frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a c^2\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{2} \left (a^2 c^2\right ) \int x \tan ^{-1}(a x) \, dx-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-2 i c^2 \tan ^{-1}(a x)^2-\frac {9}{4} a c^2 x \tan ^{-1}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} c^2 \tan ^{-1}(a x)^3+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-6 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{2} \left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\left (6 a c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^3 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx\\ &=-\frac {1}{4} a c^2 x+\frac {1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-2 i c^2 \tan ^{-1}(a x)^2-\frac {9}{4} a c^2 x \tan ^{-1}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} c^2 \tan ^{-1}(a x)^3+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )-\left (6 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{4} \left (a c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {1}{4} a c^2 x+\frac {1}{4} c^2 \tan ^{-1}(a x)+\frac {1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-2 i c^2 \tan ^{-1}(a x)^2-\frac {9}{4} a c^2 x \tan ^{-1}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} c^2 \tan ^{-1}(a x)^3+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-3 i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{2} \left (3 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )\\ &=-\frac {1}{4} a c^2 x+\frac {1}{4} c^2 \tan ^{-1}(a x)+\frac {1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-2 i c^2 \tan ^{-1}(a x)^2-\frac {9}{4} a c^2 x \tan ^{-1}(a x)^2-\frac {1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+\frac {3}{4} c^2 \tan ^{-1}(a x)^3+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac {1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-2 i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 302, normalized size = 0.82 \begin {gather*} \frac {1}{64} c^2 \left (-i \pi ^4-16 a x+16 \text {ArcTan}(a x)+16 a^2 x^2 \text {ArcTan}(a x)+128 i \text {ArcTan}(a x)^2-144 a x \text {ArcTan}(a x)^2-16 a^3 x^3 \text {ArcTan}(a x)^2+48 \text {ArcTan}(a x)^3+64 a^2 x^2 \text {ArcTan}(a x)^3+16 a^4 x^4 \text {ArcTan}(a x)^3+32 i \text {ArcTan}(a x)^4+64 \text {ArcTan}(a x)^3 \log \left (1-e^{-2 i \text {ArcTan}(a x)}\right )-256 \text {ArcTan}(a x) \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )-64 \text {ArcTan}(a x)^3 \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )+96 i \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(a x)}\right )+32 i \left (4+3 \text {ArcTan}(a x)^2\right ) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )+96 \text {ArcTan}(a x) \text {PolyLog}\left (3,e^{-2 i \text {ArcTan}(a x)}\right )-96 \text {ArcTan}(a x) \text {PolyLog}\left (3,-e^{2 i \text {ArcTan}(a x)}\right )-48 i \text {PolyLog}\left (4,e^{-2 i \text {ArcTan}(a x)}\right )-48 i \text {PolyLog}\left (4,-e^{2 i \text {ArcTan}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 32.36, size = 566, normalized size = 1.53
method | result | size |
derivativedivides | \(\frac {c^{2} \left (-8 \arctan \left (a x \right )^{2}-3 i \arctan \left (a x \right )^{3}+i \arctan \left (a x \right )^{2} a x +3 \arctan \left (a x \right )^{3} a x -\arctan \left (a x \right )^{2} a^{2} x^{2}-i \arctan \left (a x \right )^{3} a^{2} x^{2}+\arctan \left (a x \right )^{3} a^{3} x^{3}-1-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x \right ) \left (a x +i\right )}{4}-c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c^{2} \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c^{2} \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{2} \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{2} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+6 c^{2} \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 i c^{2} \arctan \left (a x \right )^{2}+2 i c^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {3 i c^{2} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 i c^{2} \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{2} \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\) | \(566\) |
default | \(\frac {c^{2} \left (-8 \arctan \left (a x \right )^{2}-3 i \arctan \left (a x \right )^{3}+i \arctan \left (a x \right )^{2} a x +3 \arctan \left (a x \right )^{3} a x -\arctan \left (a x \right )^{2} a^{2} x^{2}-i \arctan \left (a x \right )^{3} a^{2} x^{2}+\arctan \left (a x \right )^{3} a^{3} x^{3}-1-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x \right ) \left (a x +i\right )}{4}-c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c^{2} \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c^{2} \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{2} \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{2} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+6 c^{2} \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 i c^{2} \arctan \left (a x \right )^{2}+2 i c^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {3 i c^{2} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 i c^{2} \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{2} \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\) | \(566\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{2} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 2 a^{2} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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