Optimal. Leaf size=113 \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}-\frac{x}{2 a^3 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c} \]
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Rubi [A] time = 0.142134, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4916, 4852, 321, 203, 4920, 4854, 2402, 2315} \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}-\frac{x}{2 a^3 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c} \]
Antiderivative was successfully verified.
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Rule 4916
Rule 4852
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int x \tan ^{-1}(a x) \, dx}{a^2 c}\\ &=\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^3 c}-\frac{\int \frac{x^2}{1+a^2 x^2} \, dx}{2 a c}\\ &=-\frac{x}{2 a^3 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4 c}+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{2 a^3 c}-\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c}\\ &=-\frac{x}{2 a^3 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4 c}+\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^4 c}\\ &=-\frac{x}{2 a^3 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4 c}+\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c}\\ \end{align*}
Mathematica [A] time = 0.0338757, size = 120, normalized size = 1.06 \[ \frac{i \text{PolyLog}\left (2,-\frac{a x+i}{-a x+i}\right )}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}-\frac{x}{2 a^3 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{\log \left (\frac{2 i}{-a x+i}\right ) \tan ^{-1}(a x)}{a^4 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.091, size = 238, normalized size = 2.1 \begin{align*}{\frac{{x}^{2}\arctan \left ( ax \right ) }{2\,{a}^{2}c}}-{\frac{\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{a}^{4}c}}-{\frac{x}{2\,{a}^{3}c}}+{\frac{\arctan \left ( ax \right ) }{2\,{a}^{4}c}}-{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{a}^{4}c}}+{\frac{{\frac{i}{8}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{4}c}}+{\frac{{\frac{i}{4}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}c}}+{\frac{{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}c}}+{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{a}^{4}c}}-{\frac{{\frac{i}{8}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{4}c}}-{\frac{{\frac{i}{4}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}c}}-{\frac{{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )}{a^{2} c x^{2} + c}\,{d x} \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 (\frac{x^{3} \arctan \left (a x\right )}{a^{2} c x^{2} + c}, 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*} \frac{\int \frac{x^{3} \operatorname{atan}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \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 \frac{x^{3} \arctan \left (a x\right )}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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