Optimal. Leaf size=133 \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}-\frac{x}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x)}{4 a^4 c^2}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c^2} \]
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Rubi [A] time = 0.162915, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4964, 4920, 4854, 2402, 2315, 4930, 199, 205} \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}-\frac{x}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x)}{4 a^4 c^2}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 4964
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{\int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2}+\frac{\int \frac{x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2 c}\\ &=\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^3}-\frac{\int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^3 c^2}\\ &=-\frac{x}{4 a^3 c^2 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}+\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c^2}-\frac{\int \frac{1}{c+a^2 c x^2} \, dx}{4 a^3 c}\\ &=-\frac{x}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{4 a^4 c^2}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}-\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^4 c^2}\\ &=-\frac{x}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{4 a^4 c^2}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}-\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}\\ \end{align*}
Mathematica [A] time = 0.121025, size = 77, normalized size = 0.58 \[ \frac{4 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+4 i \tan ^{-1}(a x)^2-\sin \left (2 \tan ^{-1}(a x)\right )+2 \tan ^{-1}(a x) \left (\cos \left (2 \tan ^{-1}(a x)\right )-4 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{8 a^4 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.095, size = 257, normalized size = 1.9 \begin{align*}{\frac{\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{a}^{4}{c}^{2}}}+{\frac{\arctan \left ( ax \right ) }{2\,{a}^{4}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{x}{4\,{a}^{3}{c}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{\arctan \left ( ax \right ) }{4\,{a}^{4}{c}^{2}}}+{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{a}^{4}{c}^{2}}}-{\frac{{\frac{i}{8}} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{4}{c}^{2}}}-{\frac{{\frac{i}{4}}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}{c}^{2}}}-{\frac{{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}{c}^{2}}}-{\frac{{\frac{i}{4}}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{a}^{4}{c}^{2}}}+{\frac{{\frac{i}{8}} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{4}{c}^{2}}}+{\frac{{\frac{i}{4}}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}{c}^{2}}}+{\frac{{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}{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{x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{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^{3} \arctan \left (a x\right )}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{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{x^{3} \operatorname{atan}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, 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{x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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