Optimal. Leaf size=137 \[ -\frac{1}{2} b \text{PolyLog}\left (2,1-\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{1}{2} b \text{PolyLog}\left (2,\frac{2}{1-c x^2}-1\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{1}{4} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x^2}\right )-\frac{1}{4} b^2 \text{PolyLog}\left (3,\frac{2}{1-c x^2}-1\right )+\tanh ^{-1}\left (1-\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \]
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Rubi [A] time = 0.335539, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6095, 5914, 6052, 5948, 6058, 6610} \[ -\frac{1}{2} b \text{PolyLog}\left (2,1-\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{1}{2} b \text{PolyLog}\left (2,\frac{2}{1-c x^2}-1\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{1}{4} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x^2}\right )-\frac{1}{4} b^2 \text{PolyLog}\left (3,\frac{2}{1-c x^2}-1\right )+\tanh ^{-1}\left (1-\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 6095
Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx,x,x^2\right )\\ &=\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x^2}\right )-(2 b c) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^2\right )\\ &=\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x^2}\right )+(b c) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^2\right )-(b c) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^2\right )\\ &=\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x^2}\right )-\frac{1}{2} b \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \text{Li}_2\left (1-\frac{2}{1-c x^2}\right )+\frac{1}{2} b \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \text{Li}_2\left (-1+\frac{2}{1-c x^2}\right )+\frac{1}{2} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^2\right )-\frac{1}{2} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,x^2\right )\\ &=\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x^2}\right )-\frac{1}{2} b \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \text{Li}_2\left (1-\frac{2}{1-c x^2}\right )+\frac{1}{2} b \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \text{Li}_2\left (-1+\frac{2}{1-c x^2}\right )+\frac{1}{4} b^2 \text{Li}_3\left (1-\frac{2}{1-c x^2}\right )-\frac{1}{4} b^2 \text{Li}_3\left (-1+\frac{2}{1-c x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.100625, size = 141, normalized size = 1.03 \[ \frac{1}{4} b \left (2 \text{PolyLog}\left (2,\frac{c x^2+1}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-2 \text{PolyLog}\left (2,\frac{c x^2+1}{c x^2-1}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+b \left (\text{PolyLog}\left (3,\frac{c x^2+1}{c x^2-1}\right )-\text{PolyLog}\left (3,\frac{c x^2+1}{1-c x^2}\right )\right )\right )+\tanh ^{-1}\left (\frac{2}{c x^2-1}+1\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.133, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}}{x}}\, dx \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*} a^{2} \log \left (x\right ) + \int \frac{b^{2}{\left (\log \left (c x^{2} + 1\right ) - \log \left (-c x^{2} + 1\right )\right )}^{2}}{4 \, x} + \frac{a b{\left (\log \left (c x^{2} + 1\right ) - \log \left (-c x^{2} + 1\right )\right )}}{x}\,{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{b^{2} \operatorname{artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{2}\right ) + a^{2}}{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{\left (a + b \operatorname{atanh}{\left (c x^{2} \right )}\right )^{2}}{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{{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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