Optimal. Leaf size=160 \[ -\frac{i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2}+\frac{c \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac{c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{4 a}+\frac{c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{4 a^2}-\frac{i c \tan ^{-1}(a x)^2}{2 a^2}-\frac{c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^2}-\frac{c x}{4 a}-\frac{c x \tan ^{-1}(a x)^2}{2 a} \]
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Rubi [A] time = 0.129083, antiderivative size = 160, normalized size of antiderivative = 1., 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 4930
Rule 4880
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 8
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*}
Mathematica [A] time = 0.0674966, size = 101, normalized size = 0.63 \[ \frac{c \left (2 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3-\left (a^3 x^3+3 a x-2 i\right ) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left (a^2 x^2-4 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+1\right )-a x\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.097, size = 276, normalized size = 1.7 \begin{align*}{\frac{{a}^{2}c \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}}{4}}+{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}}{2}}-{\frac{ac \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}}{4}}-{\frac{3\,cx \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,a}}+{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4\,{a}^{2}}}+{\frac{c\arctan \left ( ax \right ){x}^{2}}{4}}+{\frac{c\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{a}^{2}}}-{\frac{cx}{4\,a}}+{\frac{c\arctan \left ( ax \right ) }{4\,{a}^{2}}}+{\frac{{\frac{i}{8}}c \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{2}}}+{\frac{{\frac{i}{4}}c\ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \ln \left ( ax+i \right ) }{{a}^{2}}}-{\frac{{\frac{i}{4}}c\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}+{\frac{{\frac{i}{4}}c{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}}}-{\frac{{\frac{i}{8}}c \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{2}}}-{\frac{{\frac{i}{4}}c\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}}}-{\frac{{\frac{i}{4}}c{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}}}+{\frac{{\frac{i}{4}}c\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{3} + c x\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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