Optimal. Leaf size=72 \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^2 c} \]
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Rubi [A] time = 0.0721384, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4920, 4854, 2402, 2315} \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a c}\\ &=-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^2 c}+\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c}\\ &=-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^2 c}-\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^2 c}\\ &=-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^2 c}-\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^2 c}\\ \end{align*}
Mathematica [A] time = 0.0052534, size = 77, normalized size = 1.07 \[ -\frac{i \text{PolyLog}\left (2,\frac{a x+i}{a x-i}\right )}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\log \left (\frac{2 i}{-a x+i}\right ) \tan ^{-1}(a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 202, normalized size = 2.8 \begin{align*}{\frac{\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{a}^{2}c}}-{\frac{{\frac{i}{8}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{2}c}}+{\frac{{\frac{i}{4}}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}c}}-{\frac{{\frac{i}{4}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}c}}-{\frac{{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}c}}+{\frac{{\frac{i}{8}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{2}c}}+{\frac{{\frac{i}{4}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}c}}-{\frac{{\frac{i}{4}}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}c}}+{\frac{{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}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 \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 \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 \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 \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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