Optimal. Leaf size=117 \[ -\frac{i \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{a x}{4 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}-\frac{\tan ^{-1}(a x)}{4 c^2}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^2} \]
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Rubi [A] time = 0.18415, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {4966, 4924, 4868, 2447, 4930, 199, 205} \[ -\frac{i \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^2}-\frac{a x}{4 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}-\frac{\tan ^{-1}(a x)}{4 c^2}+\frac{\log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4924
Rule 4868
Rule 2447
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}-\frac{1}{2} a \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{i \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^2}\\ &=-\frac{a x}{4 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}+\frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{a \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}-\frac{a \int \frac{1}{c+a^2 c x^2} \, dx}{4 c}\\ &=-\frac{a x}{4 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{4 c^2}+\frac{\tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}+\frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{i \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.157162, size = 72, normalized size = 0.62 \[ -\frac{4 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+4 i \tan ^{-1}(a x)^2+\sin \left (2 \tan ^{-1}(a x)\right )-2 \tan ^{-1}(a x) \left (\cos \left (2 \tan ^{-1}(a x)\right )+4 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )}{8 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.049, size = 298, normalized size = 2.6 \begin{align*} -{\frac{\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{c}^{2}}}+{\frac{\arctan \left ( ax \right ) }{2\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{\arctan \left ( ax \right ) \ln \left ( ax \right ) }{{c}^{2}}}-{\frac{ax}{4\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{\arctan \left ( ax \right ) }{4\,{c}^{2}}}+{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{c}^{2}}}+{\frac{{\frac{i}{2}}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{{c}^{2}}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+iax \right ) }{{c}^{2}}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1-iax \right ) }{{c}^{2}}}-{\frac{{\frac{i}{2}}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{{c}^{2}}}+{\frac{{\frac{i}{8}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{c}^{2}}}-{\frac{{\frac{i}{8}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{c}^{2}}}-{\frac{{\frac{i}{4}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{2}}}+{\frac{{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{2}}}-{\frac{{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{2}}}+{\frac{{\frac{i}{4}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{2}}}-{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{c}^{2}}} \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{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x}\,{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{\arctan \left (a x\right )}{a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \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 )}{{\left (a^{2} c x^{2} + c\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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