Optimal. Leaf size=70 \[ \frac{1}{2} i a^2 c \text{PolyLog}(2,-i a x)-\frac{1}{2} i a^2 c \text{PolyLog}(2,i a x)-\frac{1}{2} a^2 c \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)}{2 x^2}-\frac{a c}{2 x} \]
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Rubi [A] time = 0.0706943, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4950, 4852, 325, 203, 4848, 2391} \[ \frac{1}{2} i a^2 c \text{PolyLog}(2,-i a x)-\frac{1}{2} i a^2 c \text{PolyLog}(2,i a x)-\frac{1}{2} a^2 c \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)}{2 x^2}-\frac{a c}{2 x} \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4852
Rule 325
Rule 203
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)}{x^3} \, dx &=c \int \frac{\tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac{\tan ^{-1}(a x)}{x} \, dx\\ &=-\frac{c \tan ^{-1}(a x)}{2 x^2}+\frac{1}{2} (a c) \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac{1}{2} \left (i a^2 c\right ) \int \frac{\log (1-i a x)}{x} \, dx-\frac{1}{2} \left (i a^2 c\right ) \int \frac{\log (1+i a x)}{x} \, dx\\ &=-\frac{a c}{2 x}-\frac{c \tan ^{-1}(a x)}{2 x^2}+\frac{1}{2} i a^2 c \text{Li}_2(-i a x)-\frac{1}{2} i a^2 c \text{Li}_2(i a x)-\frac{1}{2} \left (a^3 c\right ) \int \frac{1}{1+a^2 x^2} \, dx\\ &=-\frac{a c}{2 x}-\frac{1}{2} a^2 c \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)}{2 x^2}+\frac{1}{2} i a^2 c \text{Li}_2(-i a x)-\frac{1}{2} i a^2 c \text{Li}_2(i a x)\\ \end{align*}
Mathematica [C] time = 0.0045507, size = 74, normalized size = 1.06 \[ -\frac{a c \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-a^2 x^2\right )}{2 x}+\frac{1}{2} i a^2 c \text{PolyLog}(2,-i a x)-\frac{1}{2} i a^2 c \text{PolyLog}(2,i a x)-\frac{c \tan ^{-1}(a x)}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.041, size = 110, normalized size = 1.6 \begin{align*} -{\frac{c\arctan \left ( ax \right ) }{2\,{x}^{2}}}+{a}^{2}c\arctan \left ( ax \right ) \ln \left ( ax \right ) -{\frac{{a}^{2}c\arctan \left ( ax \right ) }{2}}-{\frac{ac}{2\,x}}+{\frac{i}{2}}{a}^{2}c\ln \left ( ax \right ) \ln \left ( 1+iax \right ) -{\frac{i}{2}}{a}^{2}c\ln \left ( ax \right ) \ln \left ( 1-iax \right ) +{\frac{i}{2}}{a}^{2}c{\it dilog} \left ( 1+iax \right ) -{\frac{i}{2}}{a}^{2}c{\it dilog} \left ( 1-iax \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63797, size = 142, normalized size = 2.03 \begin{align*} -\frac{\pi a^{2} c x^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, a^{2} c x^{2} \arctan \left (a x\right ) \log \left (x{\left | a \right |}\right ) + 2 i \, a^{2} c x^{2}{\rm Li}_2\left (i \, a x + 1\right ) - 2 i \, a^{2} c x^{2}{\rm Li}_2\left (-i \, a x + 1\right ) + 2 \, a c x - 2 \,{\left (a^{2} c x^{2}{\left (2 i \, \arctan \left (0, a\right ) - 1\right )} - c\right )} \arctan \left (a x\right )}{4 \, x^{2}} \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 )}{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*} c \left (\int \frac{\operatorname{atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac{a^{2} \operatorname{atan}{\left (a x \right )}}{x}\, 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 )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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