Optimal. Leaf size=160 \[ -\frac {i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^2}+\frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {i c \tan ^{-1}(a x)^2}{2 a^2}-\frac {c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^2}-\frac {c x}{4 a}-\frac {c x \tan ^{-1}(a x)^2}{2 a} \]
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Rubi [A] time = 0.13, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4930, 4880, 4846, 4920, 4854, 2402, 2315, 8} \[ -\frac {i c \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2}+\frac {c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {i c \tan ^{-1}(a x)^2}{2 a^2}-\frac {c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^2}-\frac {c x}{4 a}-\frac {c x \tan ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4880
Rule 4920
Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx &=\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {3 \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx}{4 a}\\ &=\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c \int 1 \, dx}{4 a}-\frac {c \int \tan ^{-1}(a x)^2 \, dx}{2 a}\\ &=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {c x \tan ^{-1}(a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}+c \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {i c \tan ^{-1}(a x)^2}{2 a^2}-\frac {c x \tan ^{-1}(a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a}\\ &=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {i c \tan ^{-1}(a x)^2}{2 a^2}-\frac {c x \tan ^{-1}(a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}+\frac {c \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a}\\ &=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {i c \tan ^{-1}(a x)^2}{2 a^2}-\frac {c x \tan ^{-1}(a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {(i c) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^2}\\ &=-\frac {c x}{4 a}+\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{4 a^2}-\frac {i c \tan ^{-1}(a x)^2}{2 a^2}-\frac {c x \tan ^{-1}(a x)^2}{2 a}-\frac {c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{4 a}+\frac {c \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac {c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}-\frac {i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 101, normalized size = 0.63 \[ \frac {c \left (-\left (a^3 x^3+3 a x-2 i\right ) \tan ^{-1}(a x)^2+\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+\tan ^{-1}(a x) \left (a^2 x^2-4 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+1\right )+2 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )-a x\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{3} + c x\right )} \arctan \left (a x\right )^{3}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 276, normalized size = 1.72 \[ \frac {a^{2} c \arctan \left (a x \right )^{3} x^{4}}{4}+\frac {c \arctan \left (a x \right )^{3} x^{2}}{2}-\frac {a c \arctan \left (a x \right )^{2} x^{3}}{4}-\frac {3 c x \arctan \left (a x \right )^{2}}{4 a}+\frac {c \arctan \left (a x \right )^{3}}{4 a^{2}}+\frac {c \arctan \left (a x \right ) x^{2}}{4}+\frac {c \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{2}}-\frac {c x}{4 a}+\frac {c \arctan \left (a x \right )}{4 a^{2}}-\frac {i c \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{4 a^{2}}+\frac {i c \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{4 a^{2}}-\frac {i c \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{4 a^{2}}+\frac {i c \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{4 a^{2}}+\frac {i c \ln \left (a x +i\right )^{2}}{8 a^{2}}-\frac {i c \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{4 a^{2}}+\frac {i c \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{4 a^{2}}-\frac {i c \ln \left (a x -i\right )^{2}}{8 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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 \[ c \left (\int x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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