Optimal. Leaf size=115 \[ -\frac{1}{2} b d \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-d \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}+\frac{1}{2} b d \text{csch}^{-1}(c x)^2-b d \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]
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Rubi [A] time = 0.290918, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {6304, 14, 5789, 6742, 264, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac{1}{2} b d \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-d \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}+\frac{1}{2} b d \text{csch}^{-1}(c x)^2-b d \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6304
Rule 14
Rule 5789
Rule 6742
Rule 264
Rule 2325
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right ) \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e}{2 x^2}+d \log (x)}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \left (-\frac{e}{2 x^2 \sqrt{1+\frac{x^2}{c^2}}}+\frac{d \log (x)}{\sqrt{1+\frac{x^2}{c^2}}}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{(b d) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=\frac{b e \sqrt{1+\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+b d \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-(b d) \operatorname{Subst}\left (\int \frac{\sinh ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b e \sqrt{1+\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+b d \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-(b d) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{b e \sqrt{1+\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{2} b d \text{csch}^{-1}(c x)^2+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+b d \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+(2 b d) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{b e \sqrt{1+\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{2} b d \text{csch}^{-1}(c x)^2+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )-b d \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+(b d) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{b e \sqrt{1+\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{2} b d \text{csch}^{-1}(c x)^2+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )-b d \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{1}{2} (b d) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c x)}\right )\\ &=\frac{b e \sqrt{1+\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{2} b d \text{csch}^{-1}(c x)^2+\frac{1}{2} e x^2 \left (a+b \text{csch}^{-1}(c x)\right )-b d \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{1}{2} b d \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.134219, size = 93, normalized size = 0.81 \[ \frac{b c d \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )+2 a c d \log (x)+a c e x^2+b e x \sqrt{\frac{1}{c^2 x^2}+1}+b c \text{csch}^{-1}(c x) \left (e x^2-2 d \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )-b c d \text{csch}^{-1}(c x)^2}{2 c} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.222, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) \left ( a+b{\rm arccsch} \left (cx\right ) \right ) }{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*} 2 \, b c^{2} d \int \frac{x \log \left (x\right )}{2 \,{\left (\sqrt{c^{2} x^{2} + 1} c^{2} x^{2} + c^{2} x^{2} + \sqrt{c^{2} x^{2} + 1} + 1\right )}}\,{d x} - \frac{1}{2} \, b e x^{2} \log \left (c\right ) - \frac{1}{2} \, b e x^{2} \log \left (x\right ) + \frac{1}{2} \, a e x^{2} - b d \log \left (c\right ) \log \left (x\right ) - \frac{1}{2} \, b d \log \left (x\right )^{2} - \frac{1}{4} \,{\left (2 \, \log \left (c^{2} x^{2} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-c^{2} x^{2}\right )\right )} b d + a d \log \left (x\right ) + \frac{1}{2} \,{\left (b e x^{2} + 2 \, b d \log \left (x\right )\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + \frac{b e{\left (2 \, \sqrt{c^{2} x^{2} + 1} - \log \left (c^{2} x^{2} + 1\right )\right )}}{4 \, c^{2}} + \frac{b e \log \left (c^{2} x^{2} + 1\right )}{4 \, c^{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 )} \operatorname{arcsch}\left (c x\right )}{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{acsch}{\left (c x \right )}\right ) \left (d + e 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{{\left (e x^{2} + d\right )}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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