Optimal. Leaf size=39 \[ \frac{1}{4} i b \text{PolyLog}\left (2,-i c x^2\right )-\frac{1}{4} i b \text{PolyLog}\left (2,i c x^2\right )+a \log (x) \]
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Rubi [A] time = 0.0454425, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5031, 4848, 2391} \[ \frac{1}{4} i b \text{PolyLog}\left (2,-i c x^2\right )-\frac{1}{4} i b \text{PolyLog}\left (2,i c x^2\right )+a \log (x) \]
Antiderivative was successfully verified.
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Rule 5031
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(c x)}{x} \, dx,x,x^2\right )\\ &=a \log (x)+\frac{1}{4} (i b) \operatorname{Subst}\left (\int \frac{\log (1-i c x)}{x} \, dx,x,x^2\right )-\frac{1}{4} (i b) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,x^2\right )\\ &=a \log (x)+\frac{1}{4} i b \text{Li}_2\left (-i c x^2\right )-\frac{1}{4} i b \text{Li}_2\left (i c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0047769, size = 39, normalized size = 1. \[ \frac{1}{4} i b \text{PolyLog}\left (2,-i c x^2\right )-\frac{1}{4} i b \text{PolyLog}\left (2,i c x^2\right )+a \log (x) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.082, size = 63, normalized size = 1.6 \begin{align*} a\ln \left ( x \right ) +b\ln \left ( x \right ) \arctan \left ( c{x}^{2} \right ) -{\frac{b}{2\,c}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}{{\it \_Z}}^{4}+1 \right ) }{\frac{1}{{{\it \_R1}}^{2}} \left ( \ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) \right ) }} \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*} b \int \frac{\arctan \left (c x^{2}\right )}{x}\,{d x} + a \log \left (x\right ) \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{b \arctan \left (c x^{2}\right ) + a}{x}, 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{a + b \operatorname{atan}{\left (c x^{2} \right )}}{x}\, 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{b \arctan \left (c x^{2}\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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