Optimal. Leaf size=77 \[ \frac{1}{2} i b e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e \text{PolyLog}(2,i c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e \log (x)-\frac{1}{2} b c^2 d \tan ^{-1}(c x)-\frac{b c d}{2 x} \]
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Rubi [A] time = 0.0991435, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {4980, 4852, 325, 203, 4848, 2391} \[ \frac{1}{2} i b e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e \text{PolyLog}(2,i c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e \log (x)-\frac{1}{2} b c^2 d \tan ^{-1}(c x)-\frac{b c d}{2 x} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4852
Rule 325
Rule 203
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac{e \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+e \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e \log (x)+\frac{1}{2} (b c d) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac{1}{2} (i b e) \int \frac{\log (1-i c x)}{x} \, dx-\frac{1}{2} (i b e) \int \frac{\log (1+i c x)}{x} \, dx\\ &=-\frac{b c d}{2 x}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e \log (x)+\frac{1}{2} i b e \text{Li}_2(-i c x)-\frac{1}{2} i b e \text{Li}_2(i c x)-\frac{1}{2} \left (b c^3 d\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{b c d}{2 x}-\frac{1}{2} b c^2 d \tan ^{-1}(c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e \log (x)+\frac{1}{2} i b e \text{Li}_2(-i c x)-\frac{1}{2} i b e \text{Li}_2(i c x)\\ \end{align*}
Mathematica [C] time = 0.0047646, size = 86, normalized size = 1.12 \[ -\frac{b c d \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )}{2 x}+\frac{1}{2} i b e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e \text{PolyLog}(2,i c x)-\frac{a d}{2 x^2}+a e \log (x)-\frac{b d \tan ^{-1}(c x)}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.053, size = 117, normalized size = 1.5 \begin{align*} -{\frac{ad}{2\,{x}^{2}}}+ae\ln \left ( cx \right ) -{\frac{\arctan \left ( cx \right ) bd}{2\,{x}^{2}}}+b\arctan \left ( cx \right ) e\ln \left ( cx \right ) +{\frac{i}{2}}be\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -{\frac{i}{2}}be\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +{\frac{i}{2}}be{\it dilog} \left ( 1+icx \right ) -{\frac{i}{2}}be{\it dilog} \left ( 1-icx \right ) -{\frac{b{c}^{2}d\arctan \left ( cx \right ) }{2}}-{\frac{bcd}{2\,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*} -\frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b d + b e \int \frac{\arctan \left (c x\right )}{x}\,{d x} + a e \log \left (x\right ) - \frac{a d}{2 \, 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{a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \arctan \left (c 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*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \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 (e x^{2} + d\right )}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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