Optimal. Leaf size=135 \[ -\frac{2}{3} i a^3 c \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{a^2 c}{3 x}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c \tan ^{-1}(a x)-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{c \tan ^{-1}(a x)^2}{3 x^3} \]
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Rubi [A] time = 0.312138, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4950, 4852, 4918, 325, 203, 4924, 4868, 2447} \[ -\frac{2}{3} i a^3 c \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{a^2 c}{3 x}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c \tan ^{-1}(a x)-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{c \tan ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4852
Rule 4918
Rule 325
Rule 203
Rule 4924
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2}{x^4} \, dx &=c \int \frac{\tan ^{-1}(a x)^2}{x^4} \, dx+\left (a^2 c\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^2}{3 x^3}-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{1}{3} (2 a c) \int \frac{\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (2 a^3 c\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-i a^3 c \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{3 x^3}-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{1}{3} (2 a c) \int \frac{\tan ^{-1}(a x)}{x^3} \, dx+\left (2 i a^3 c\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\frac{1}{3} \left (2 a^3 c\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{3 x^3}-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+2 a^3 c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )+\frac{1}{3} \left (a^2 c\right ) \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 i a^3 c\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\left (2 a^4 c\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c}{3 x}-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{3 x^3}-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a^3 c \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{1}{3} \left (a^4 c\right ) \int \frac{1}{1+a^2 x^2} \, dx+\frac{1}{3} \left (2 a^4 c\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c}{3 x}-\frac{1}{3} a^3 c \tan ^{-1}(a x)-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{3 x^3}-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{2}{3} i a^3 c \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.590859, size = 103, normalized size = 0.76 \[ \frac{c \left (-2 i a^3 x^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )-a^2 x^2+a x \tan ^{-1}(a x) \left (-a^2 x^2+4 a^2 x^2 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )-1\right )+(1-2 i a x) (a x-i)^2 \tan ^{-1}(a x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.096, size = 323, normalized size = 2.4 \begin{align*} -{\frac{{a}^{2}c \left ( \arctan \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{2\,{a}^{3}c\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{ac\arctan \left ( ax \right ) }{3\,{x}^{2}}}+{\frac{4\,{a}^{3}c\arctan \left ( ax \right ) \ln \left ( ax \right ) }{3}}-{\frac{{a}^{3}c\arctan \left ( ax \right ) }{3}}-{\frac{{a}^{2}c}{3\,x}}-{\frac{i}{3}}{a}^{3}c\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{i}{3}}{a}^{3}c{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{i}{3}}{a}^{3}c\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) -{\frac{i}{3}}{a}^{3}c{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) -{\frac{2\,i}{3}}{a}^{3}c{\it dilog} \left ( 1-iax \right ) +{\frac{2\,i}{3}}{a}^{3}c{\it dilog} \left ( 1+iax \right ) +{\frac{2\,i}{3}}{a}^{3}c\ln \left ( ax \right ) \ln \left ( 1+iax \right ) -{\frac{2\,i}{3}}{a}^{3}c\ln \left ( ax \right ) \ln \left ( 1-iax \right ) +{\frac{i}{3}}{a}^{3}c\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) +{\frac{i}{6}}{a}^{3}c \left ( \ln \left ( ax-i \right ) \right ) ^{2}+{\frac{i}{3}}{a}^{3}c\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) -{\frac{i}{6}}{a}^{3}c \left ( \ln \left ( ax+i \right ) \right ) ^{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{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{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*} c \left (\int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{4}}\, dx + \int \frac{a^{2} \operatorname{atan}^{2}{\left (a x \right )}}{x^{2}}\, dx\right ) \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{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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