Optimal. Leaf size=205 \[ \frac{3 i a^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{19 a^3 x}{32 c^3 \left (a^2 x^2+1\right )}+\frac{a^3 x}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2 \tan ^{-1}(a x)}{c^3 \left (a^2 x^2+1\right )}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{3 a^2 \tan ^{-1}(a x)}{32 c^3}-\frac{3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^3}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a}{2 c^3 x} \]
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Rubi [A] time = 0.760318, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4930, 199, 205} \[ \frac{3 i a^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{2 c^3}+\frac{19 a^3 x}{32 c^3 \left (a^2 x^2+1\right )}+\frac{a^3 x}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2 \tan ^{-1}(a x)}{c^3 \left (a^2 x^2+1\right )}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{3 a^2 \tan ^{-1}(a x)}{32 c^3}-\frac{3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^3}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a}{2 c^3 x} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4918
Rule 4852
Rule 325
Rule 203
Rule 4924
Rule 4868
Rule 2447
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{x^3 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{4} a^3 \int \frac{1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)}{x^3} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \left (\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{a^4 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac{a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{a \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^3}-\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}+\frac{\left (3 a^3\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (\frac{a^2 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{\left (i a^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}-\frac{a^3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=-\frac{a}{2 c^3 x}+\frac{a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x}{32 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac{a^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{a^3 \int \frac{1}{1+a^2 x^2} \, dx}{2 c^3}+\frac{a^3 \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac{\left (3 a^3\right ) \int \frac{1}{c+a^2 c x^2} \, dx}{32 c^2}-2 \left (-\frac{a^3 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac{a^2 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{a^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{a^3 \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-\frac{a^3 \int \frac{1}{c+a^2 c x^2} \, dx}{4 c^2}\right )\\ &=-\frac{a}{2 c^3 x}+\frac{a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 x}{32 c^3 \left (1+a^2 x^2\right )}-\frac{13 a^2 \tan ^{-1}(a x)}{32 c^3}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac{a^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{i a^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac{a^3 x}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3}+\frac{a^2 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{a^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )\\ \end{align*}
Mathematica [A] time = 0.592833, size = 111, normalized size = 0.54 \[ \frac{a^2 \left (192 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+\tan ^{-1}(a x) \left (-\frac{64}{a^2 x^2}-384 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )-80 \cos \left (2 \tan ^{-1}(a x)\right )-4 \cos \left (4 \tan ^{-1}(a x)\right )-64\right )-\frac{64}{a x}+192 i \tan ^{-1}(a x)^2+40 \sin \left (2 \tan ^{-1}(a x)\right )+\sin \left (4 \tan ^{-1}(a x)\right )\right )}{128 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.103, size = 415, normalized size = 2. \begin{align*} -{\frac{{a}^{2}\arctan \left ( ax \right ) }{4\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,{a}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{c}^{3}}}-{\frac{{a}^{2}\arctan \left ( ax \right ) }{{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{\arctan \left ( ax \right ) }{2\,{c}^{3}{x}^{2}}}-3\,{\frac{{a}^{2}\arctan \left ( ax \right ) \ln \left ( ax \right ) }{{c}^{3}}}+{\frac{{\frac{3\,i}{4}}{a}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{3}}}-{\frac{{\frac{3\,i}{2}}{a}^{2}{\it dilog} \left ( 1+iax \right ) }{{c}^{3}}}-{\frac{{\frac{3\,i}{8}}{a}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{c}^{3}}}-{\frac{{\frac{3\,i}{4}}{a}^{2}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{c}^{3}}}+{\frac{{\frac{3\,i}{8}}{a}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{c}^{3}}}+{\frac{{\frac{3\,i}{2}}{a}^{2}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) }{{c}^{3}}}+{\frac{{\frac{3\,i}{2}}{a}^{2}{\it dilog} \left ( 1-iax \right ) }{{c}^{3}}}-{\frac{{\frac{3\,i}{2}}{a}^{2}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) }{{c}^{3}}}+{\frac{{\frac{3\,i}{4}}{a}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{c}^{3}}}-{\frac{{\frac{3\,i}{4}}{a}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{3}}}-{\frac{{\frac{3\,i}{4}}{a}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{c}^{3}}}+{\frac{{\frac{3\,i}{4}}{a}^{2}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{c}^{3}}}+{\frac{19\,{a}^{5}{x}^{3}}{32\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{21\,{a}^{3}x}{32\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,{a}^{2}\arctan \left ( ax \right ) }{32\,{c}^{3}}}-{\frac{a}{2\,{c}^{3}x}} \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 )}^{3} x^{3}}\,{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^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}, 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}{\left (a x \right )}}{a^{6} x^{9} + 3 a^{4} x^{7} + 3 a^{2} x^{5} + x^{3}}\, 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 )}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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