Optimal. Leaf size=276 \[ -\frac {3}{2} i c \text {ArcTan}(a x)^2-\frac {3}{2} a c x \text {ArcTan}(a x)^2+\frac {1}{2} c \text {ArcTan}(a x)^3+\frac {1}{2} a^2 c x^2 \text {ArcTan}(a x)^3+2 c \text {ArcTan}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-3 c \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \text {ArcTan}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \text {ArcTan}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \text {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
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Rubi [A]
time = 0.37, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used =
{5070, 4942, 5108, 5004, 5114, 5118, 6745, 4946, 5036, 4930, 5040, 4964, 2449, 2352}
\begin {gather*} \frac {1}{2} a^2 c x^2 \text {ArcTan}(a x)^3-\frac {3}{2} i c \text {ArcTan}(a x)^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+\frac {3}{2} i c \text {ArcTan}(a x)^2 \text {Li}_2\left (\frac {2}{i a x+1}-1\right )-\frac {3}{2} c \text {ArcTan}(a x) \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+\frac {3}{2} c \text {ArcTan}(a x) \text {Li}_3\left (\frac {2}{i a x+1}-1\right )+\frac {1}{2} c \text {ArcTan}(a x)^3-\frac {3}{2} i c \text {ArcTan}(a x)^2-\frac {3}{2} a c x \text {ArcTan}(a x)^2-3 c \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )+2 c \text {ArcTan}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c \text {Li}_4\left (1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c \text {Li}_4\left (\frac {2}{i a x+1}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 4930
Rule 4942
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rule 5070
Rule 5108
Rule 5114
Rule 5118
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3}{x} \, dx &=c \int \frac {\tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-(6 a c) \int \frac {\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {1}{2} (3 a c) \int \tan ^{-1}(a x)^2 \, dx+\frac {1}{2} (3 a c) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+(3 a c) \int \frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 a c) \int \frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} a c x \tan ^{-1}(a x)^2+\frac {1}{2} c \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+(3 i a c) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 i a c) \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 a^2 c\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} i c \tan ^{-1}(a x)^2-\frac {3}{2} a c x \tan ^{-1}(a x)^2+\frac {1}{2} c \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} (3 a c) \int \frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} (3 a c) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 a c) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx\\ &=-\frac {3}{2} i c \tan ^{-1}(a x)^2-\frac {3}{2} a c x \tan ^{-1}(a x)^2+\frac {1}{2} c \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-3 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+(3 a c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} i c \tan ^{-1}(a x)^2-\frac {3}{2} a c x \tan ^{-1}(a x)^2+\frac {1}{2} c \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-3 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )-(3 i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )\\ &=-\frac {3}{2} i c \tan ^{-1}(a x)^2-\frac {3}{2} a c x \tan ^{-1}(a x)^2+\frac {1}{2} c \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-3 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 284, normalized size = 1.03 \begin {gather*} \frac {1}{2} c \left (1+a^2 x^2\right ) \text {ArcTan}(a x)^3+2 c \text {ArcTan}(a x)^3 \tanh ^{-1}\left (1-\frac {2 i}{i-a x}\right )-\frac {3}{2} c \left (-i \text {ArcTan}(a x)^2+a x \text {ArcTan}(a x)^2+2 \text {ArcTan}(a x) \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )-i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )\right )+\frac {3}{2} i c \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,\frac {-i-a x}{-i+a x}\right )-\frac {3}{2} i c \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,\frac {i+a x}{-i+a x}\right )+\frac {3}{2} c \text {ArcTan}(a x) \text {PolyLog}\left (3,\frac {-i-a x}{-i+a x}\right )-\frac {3}{2} c \text {ArcTan}(a x) \text {PolyLog}\left (3,\frac {i+a x}{-i+a x}\right )-\frac {3}{4} i c \text {PolyLog}\left (4,\frac {-i-a x}{-i+a x}\right )+\frac {3}{4} i c \text {PolyLog}\left (4,\frac {i+a x}{-i+a x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 16.67, size = 460, normalized size = 1.67
method | result | size |
derivativedivides | \(\frac {c \arctan \left (a x \right )^{2} \left (-3-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x \right ) \left (a x +i\right )}{2}+6 i c \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-\frac {3 i c \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 i c \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 c \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 c \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i c \arctan \left (a x \right )^{2}\) | \(460\) |
default | \(\frac {c \arctan \left (a x \right )^{2} \left (-3-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x \right ) \left (a x +i\right )}{2}+6 i c \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-\frac {3 i c \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 i c \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 c \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 c \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i c \arctan \left (a x \right )^{2}\) | \(460\) |
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 \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int a^{2} x \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 )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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