Optimal. Leaf size=166 \[ \frac{4 i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{3 c}-\frac{a^2}{3 c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{a^3 \tan ^{-1}(a x)}{3 c}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}-\frac{8 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c}-\frac{a \tan ^{-1}(a x)}{3 c x^2}-\frac{\tan ^{-1}(a x)^2}{3 c x^3} \]
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Rubi [A] time = 0.435952, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4918, 4852, 325, 203, 4924, 4868, 2447, 4884} \[ \frac{4 i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{3 c}-\frac{a^2}{3 c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{a^3 \tan ^{-1}(a x)}{3 c}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}-\frac{8 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c}-\frac{a \tan ^{-1}(a x)}{3 c x^2}-\frac{\tan ^{-1}(a x)^2}{3 c x^3} \]
Antiderivative was successfully verified.
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Rule 4918
Rule 4852
Rule 325
Rule 203
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^4} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^2}{3 c x^3}+a^4 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^2}{3 c x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x^3} \, dx}{3 c}-\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac{a \tan ^{-1}(a x)}{3 c x^2}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{\tan ^{-1}(a x)^2}{3 c x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{a^2 \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c}-\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{3 c}-\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c}\\ &=-\frac{a^2}{3 c x}-\frac{a \tan ^{-1}(a x)}{3 c x^2}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{\tan ^{-1}(a x)^2}{3 c x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}-\frac{8 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{3 c}-\frac{a^4 \int \frac{1}{1+a^2 x^2} \, dx}{3 c}+\frac{\left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c}+\frac{\left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac{a^2}{3 c x}-\frac{a^3 \tan ^{-1}(a x)}{3 c}-\frac{a \tan ^{-1}(a x)}{3 c x^2}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{\tan ^{-1}(a x)^2}{3 c x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}-\frac{8 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{3 c}+\frac{4 i a^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{3 c}\\ \end{align*}
Mathematica [A] time = 0.345505, size = 120, normalized size = 0.72 \[ \frac{a^3 \left (4 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )-\frac{\frac{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{a^2 x^2}-4 \tan ^{-1}(a x)^2+1}{a x}+\tan ^{-1}(a x) \left (-\frac{a^2 x^2+1}{a^2 x^2}+\tan ^{-1}(a x) \left (\tan ^{-1}(a x)+4 i\right )-8 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )\right )}{3 c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.108, size = 374, normalized size = 2.3 \begin{align*}{\frac{{a}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{3\,c}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,c{x}^{3}}}+{\frac{{a}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{cx}}+{\frac{4\,{a}^{3}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,c}}-{\frac{a\arctan \left ( ax \right ) }{3\,c{x}^{2}}}-{\frac{8\,{a}^{3}\arctan \left ( ax \right ) \ln \left ( ax \right ) }{3\,c}}-{\frac{{a}^{3}\arctan \left ( ax \right ) }{3\,c}}-{\frac{{a}^{2}}{3\,cx}}+{\frac{{\frac{4\,i}{3}}{a}^{3}{\it dilog} \left ( 1-iax \right ) }{c}}-{\frac{{\frac{2\,i}{3}}{a}^{3}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{c}}+{\frac{{\frac{2\,i}{3}}{a}^{3}\ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \ln \left ( ax+i \right ) }{c}}-{\frac{{\frac{i}{3}}{a}^{3} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{c}}-{\frac{{\frac{4\,i}{3}}{a}^{3}{\it dilog} \left ( 1+iax \right ) }{c}}-{\frac{{\frac{4\,i}{3}}{a}^{3}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{c}}+{\frac{{\frac{2\,i}{3}}{a}^{3}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{c}}-{\frac{{\frac{2\,i}{3}}{a}^{3}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{c}}+{\frac{{\frac{4\,i}{3}}{a}^{3}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{c}}-{\frac{{\frac{2\,i}{3}}{a}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{c}}+{\frac{{\frac{2\,i}{3}}{a}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{c}}+{\frac{{\frac{i}{3}}{a}^{3} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{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^{6} + c x^{4}}, 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^{6} + x^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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