Optimal. Leaf size=250 \[ -\frac{i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{31 a^2 x}{64 c^3 \left (a^2 x^2+1\right )}+\frac{a^2 x}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{i a \tan ^{-1}(a x)^2}{c^3}+\frac{31 a \tan ^{-1}(a x)}{64 c^3}+\frac{2 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^3} \]
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Rubi [A] time = 0.551444, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4966, 4918, 4852, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205, 4900} \[ -\frac{i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{31 a^2 x}{64 c^3 \left (a^2 x^2+1\right )}+\frac{a^2 x}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{i a \tan ^{-1}(a x)^2}{c^3}+\frac{31 a \tan ^{-1}(a x)}{64 c^3}+\frac{2 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^3} \]
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
Rule 4900
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{8} a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{\left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac{a^2 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}-\frac{7 a \tan ^{-1}(a x)^3}{24 c^3}+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c^2}+\frac{\left (3 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}+\frac{\left (3 a^3\right ) \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}+\frac{a^3 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac{a^2 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}+\frac{\left (3 a^2\right ) \int \frac{1}{c+a^2 c x^2} \, dx}{64 c^2}+\frac{\left (3 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac{a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=\frac{a^2 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{31 a^2 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac{3 a \tan ^{-1}(a x)}{64 c^3}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}+\frac{(2 i a) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}+\frac{\left (3 a^2\right ) \int \frac{1}{c+a^2 c x^2} \, dx}{16 c^2}+\frac{a^2 \int \frac{1}{c+a^2 c x^2} \, dx}{4 c^2}\\ &=\frac{a^2 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{31 a^2 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac{31 a \tan ^{-1}(a x)}{64 c^3}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}+\frac{2 a \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\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^3}\\ &=\frac{a^2 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{31 a^2 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac{31 a \tan ^{-1}(a x)}{64 c^3}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}+\frac{2 a \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}\\ \end{align*}
Mathematica [A] time = 0.400255, size = 139, normalized size = 0.56 \[ -\frac{256 i a x \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+160 a x \tan ^{-1}(a x)^3+8 \tan ^{-1}(a x)^2 \left (32 i a x+16 a x \sin \left (2 \tan ^{-1}(a x)\right )+a x \sin \left (4 \tan ^{-1}(a x)\right )+32\right )-a x \left (64 \sin \left (2 \tan ^{-1}(a x)\right )+\sin \left (4 \tan ^{-1}(a x)\right )\right )+4 a x \tan ^{-1}(a x) \left (-128 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+32 \cos \left (2 \tan ^{-1}(a x)\right )+\cos \left (4 \tan ^{-1}(a x)\right )\right )}{256 c^3 x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.115, size = 440, normalized size = 1.8 \begin{align*} -{\frac{7\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{a}^{4}{x}^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{9\,{a}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{5\,a \left ( \arctan \left ( ax \right ) \right ) ^{3}}{8\,{c}^{3}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{{c}^{3}x}}-{\frac{a\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{c}^{3}}}-{\frac{a\arctan \left ( ax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{7\,a\arctan \left ( ax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{a\arctan \left ( ax \right ) \ln \left ( ax \right ) }{{c}^{3}}}-{\frac{{\frac{i}{4}}a \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{c}^{3}}}+{\frac{{\frac{i}{4}}a \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{c}^{3}}}+{\frac{{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{c}^{3}}}+{\frac{ia\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{{c}^{3}}}-{\frac{{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{3}}}-{\frac{ia\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{{c}^{3}}}-{\frac{{\frac{i}{2}}a{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{3}}}-{\frac{ia{\it dilog} \left ( 1-iax \right ) }{{c}^{3}}}+{\frac{{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{3}}}+{\frac{ia{\it dilog} \left ( 1+iax \right ) }{{c}^{3}}}-{\frac{{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{c}^{3}}}+{\frac{{\frac{i}{2}}a{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{3}}}+{\frac{31\,{x}^{3}{a}^{4}}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{33\,{a}^{2}x}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{31\,a\arctan \left ( ax \right ) }{64\,{c}^{3}}} \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^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} 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^{6} x^{8} + 3 a^{4} x^{6} + 3 a^{2} x^{4} + x^{2}}\, dx}{c^{3}} \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 )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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