Optimal. Leaf size=177 \[ -\frac{i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{a^2 x}{4 c^2 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{i a \tan ^{-1}(a x)^2}{c^2}+\frac{a \tan ^{-1}(a x)}{4 c^2}+\frac{2 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^2} \]
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Rubi [A] time = 0.339919, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4966, 4918, 4852, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205} \[ -\frac{i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{a^2 x}{4 c^2 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{i a \tan ^{-1}(a x)^2}{c^2}+\frac{a \tan ^{-1}(a x)}{4 c^2}+\frac{2 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4918
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rule 4892
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)^3}{6 c^2}+a^3 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c^2}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c}\\ &=-\frac{a \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}+\frac{1}{2} a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\\ &=\frac{a^2 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^2}{c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}+\frac{(2 i a) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^2}+\frac{a^2 \int \frac{1}{c+a^2 c x^2} \, dx}{4 c}\\ &=\frac{a^2 x}{4 c^2 \left (1+a^2 x^2\right )}+\frac{a \tan ^{-1}(a x)}{4 c^2}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^2}{c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}+\frac{2 a \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\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^2}\\ &=\frac{a^2 x}{4 c^2 \left (1+a^2 x^2\right )}+\frac{a \tan ^{-1}(a x)}{4 c^2}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^2}{c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}+\frac{2 a \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{i a \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}\\ \end{align*}
Mathematica [A] time = 0.325082, size = 109, normalized size = 0.62 \[ -\frac{8 i a x \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+4 a x \tan ^{-1}(a x)^3+2 \tan ^{-1}(a x)^2 \left (4 i a x+a x \sin \left (2 \tan ^{-1}(a x)\right )+4\right )-a x \sin \left (2 \tan ^{-1}(a x)\right )+2 a x \tan ^{-1}(a x) \left (\cos \left (2 \tan ^{-1}(a x)\right )-8 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )}{8 c^2 x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.11, size = 369, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{a \left ( \arctan \left ( ax \right ) \right ) ^{3}}{2\,{c}^{2}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{{c}^{2}x}}-{\frac{a\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}-{\frac{a\arctan \left ( ax \right ) }{2\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{a\arctan \left ( ax \right ) \ln \left ( ax \right ) }{{c}^{2}}}+{\frac{{a}^{2}x}{4\,{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{a\arctan \left ( ax \right ) }{4\,{c}^{2}}}-{\frac{{\frac{i}{2}}a\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{c}^{2}}}-{\frac{{\frac{i}{4}}a \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{c}^{2}}}+{\frac{ia\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{{c}^{2}}}+{\frac{{\frac{i}{2}}a\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{c}^{2}}}-{\frac{ia{\it dilog} \left ( 1-iax \right ) }{{c}^{2}}}+{\frac{{\frac{i}{4}}a \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{c}^{2}}}-{\frac{ia\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{{c}^{2}}}+{\frac{ia{\it dilog} \left ( 1+iax \right ) }{{c}^{2}}}-{\frac{{\frac{i}{2}}a{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{2}}}+{\frac{{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{2}}}-{\frac{{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{2}}}+{\frac{{\frac{i}{2}}a{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{2}}} \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^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} 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^{4} x^{6} + 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}} \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 )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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