Optimal. Leaf size=92 \[ -\frac{i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c}-\frac{a \tan ^{-1}(a x)^3}{3 c}-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}+\frac{2 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c} \]
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Rubi [A] time = 0.195493, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4918, 4852, 4924, 4868, 2447, 4884} \[ -\frac{i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c}-\frac{a \tan ^{-1}(a x)^3}{3 c}-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}+\frac{2 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c} \]
Antiderivative was successfully verified.
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Rule 4918
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^2}{c x}-\frac{a \tan ^{-1}(a x)^3}{3 c}+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}-\frac{a \tan ^{-1}(a x)^3}{3 c}+\frac{(2 i a) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c}\\ &=-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}-\frac{a \tan ^{-1}(a x)^3}{3 c}+\frac{2 a \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c}-\frac{\left (2 a^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}-\frac{a \tan ^{-1}(a x)^3}{3 c}+\frac{2 a \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c}-\frac{i a \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.179234, size = 73, normalized size = 0.79 \[ \frac{a \left (-i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )-\frac{1}{3} \tan ^{-1}(a x) \left (\left (\tan ^{-1}(a x)+3 i\right ) \tan ^{-1}(a x)+\frac{3 \tan ^{-1}(a x)}{a x}-6 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )\right )}{c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.099, size = 292, normalized size = 3.2 \begin{align*} -{\frac{a \left ( \arctan \left ( ax \right ) \right ) ^{3}}{3\,c}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{cx}}-{\frac{a\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{c}}+2\,{\frac{a\arctan \left ( ax \right ) \ln \left ( ax \right ) }{c}}+{\frac{{\frac{i}{2}}a{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{c}}+{\frac{ia{\it dilog} \left ( 1+iax \right ) }{c}}-{\frac{{\frac{i}{2}}a{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{c}}-{\frac{{\frac{i}{4}}a \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{c}}+{\frac{{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{c}}-{\frac{{\frac{i}{2}}a\ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \ln \left ( ax+i \right ) }{c}}-{\frac{{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{c}}+{\frac{ia\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{c}}+{\frac{{\frac{i}{4}}a \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{c}}-{\frac{ia{\it dilog} \left ( 1-iax \right ) }{c}}+{\frac{{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{c}}-{\frac{ia\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{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{\arctan \left (a x\right )^{2}}{a^{2} c x^{4} + c x^{2}}, 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{\operatorname{atan}^{2}{\left (a x \right )}}{a^{2} x^{4} + x^{2}}\, 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{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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