Optimal. Leaf size=98 \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^3 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}+\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^3 c} \]
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Rubi [A] time = 0.166925, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4916, 4846, 4920, 4854, 2402, 2315, 4884} \[ \frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^3 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}+\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^3 c} \]
Antiderivative was successfully verified.
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Rule 4916
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4884
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int \tan ^{-1}(a x)^2 \, dx}{a^2 c}\\ &=\frac{x \tan ^{-1}(a x)^2}{a^2 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}-\frac{2 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a c}\\ &=\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{2 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^2 c}\\ &=\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^3 c}-\frac{2 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2 c}\\ &=\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^3 c}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^3 c}\\ &=\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^3 c}+\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^3 c}\\ \end{align*}
Mathematica [A] time = 0.172592, size = 69, normalized size = 0.7 \[ \frac{-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-\frac{1}{3} \tan ^{-1}(a x) \left (\tan ^{-1}(a x)^2+(-3 a x+3 i) \tan ^{-1}(a x)-6 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{a^3 c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.093, size = 230, normalized size = 2.4 \begin{align*}{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{{a}^{2}c}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{3\,{a}^{3}c}}-{\frac{\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}c}}-{\frac{{\frac{i}{2}}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}c}}+{\frac{{\frac{i}{4}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}c}}+{\frac{{\frac{i}{2}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}c}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}c}}+{\frac{{\frac{i}{2}}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}c}}-{\frac{{\frac{i}{4}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}c}}-{\frac{{\frac{i}{2}}\ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \ln \left ( ax+i \right ) }{{a}^{3}c}}-{\frac{{\frac{i}{2}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}c}} \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 (\frac{x^{2} \arctan \left (a x\right )^{2}}{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^{2} \operatorname{atan}^{2}{\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^{2} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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