Optimal. Leaf size=178 \[ -\frac{1}{2} b d^2 \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-d^2 \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (6 c^2 d-e\right )}{6 c^3}+\frac{b e^2 x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{12 c}+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2-b d^2 \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d^2 \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]
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Rubi [A] time = 0.421449, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {6304, 266, 43, 5789, 6742, 453, 264, 2325, 5659, 3716, 2190, 2279, 2391} \[ -\frac{1}{2} b d^2 \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )-d^2 \log \left (\frac{1}{x}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (6 c^2 d-e\right )}{6 c^3}+\frac{b e^2 x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{12 c}+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2-b d^2 \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d^2 \log \left (\frac{1}{x}\right ) \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6304
Rule 266
Rule 43
Rule 5789
Rule 6742
Rule 453
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 )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right )^2 \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \left (-\frac{e \left (e+4 d x^2\right )}{4 x^4 \sqrt{1+\frac{x^2}{c^2}}}+\frac{d^2 \log (x)}{\sqrt{1+\frac{x^2}{c^2}}}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b d^2\right ) \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{e+4 d x^2}{x^4 \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+b d^2 \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\sinh ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )-\frac{\left (b \left (6 c^2 d-e\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c^3}\\ &=\frac{b \left (6 c^2 d-e\right ) e \sqrt{1+\frac{1}{c^2 x^2}} x}{6 c^3}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+b d^2 \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\left (b d^2\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{b \left (6 c^2 d-e\right ) e \sqrt{1+\frac{1}{c^2 x^2}} x}{6 c^3}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^3}{12 c}+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+b d^2 \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\left (2 b d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{b \left (6 c^2 d-e\right ) e \sqrt{1+\frac{1}{c^2 x^2}} x}{6 c^3}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^3}{12 c}+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )-b d^2 \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d^2 \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\left (b d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{b \left (6 c^2 d-e\right ) e \sqrt{1+\frac{1}{c^2 x^2}} x}{6 c^3}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^3}{12 c}+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )-b d^2 \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d^2 \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{1}{2} \left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c x)}\right )\\ &=\frac{b \left (6 c^2 d-e\right ) e \sqrt{1+\frac{1}{c^2 x^2}} x}{6 c^3}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^3}{12 c}+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2+d e x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{4} e^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )-b d^2 \text{csch}^{-1}(c x) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+b d^2 \text{csch}^{-1}(c x) \log \left (\frac{1}{x}\right )-d^2 \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{1}{2} b d^2 \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.43977, size = 148, normalized size = 0.83 \[ \frac{1}{2} b d^2 \left (\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\text{csch}^{-1}(c x) \left (\text{csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )\right )+a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4+\frac{b d e x \left (\sqrt{\frac{1}{c^2 x^2}+1}+c x \text{csch}^{-1}(c x)\right )}{c}+\frac{b e^2 x \left (\sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 x^2-2\right )+3 c^3 x^3 \text{csch}^{-1}(c x)\right )}{12 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.383, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e{x}^{2}+d \right ) ^{2} \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*} \frac{1}{4} \, a e^{2} x^{4} + 4 \, b c^{2} d^{2} \int \frac{x \log \left (x\right )}{4 \,{\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} + a d e x^{2} - b d^{2} \log \left (c\right ) \log \left (x\right ) - \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^{2} + a d^{2} \log \left (x\right ) + \frac{b d e{\left (2 \, \sqrt{c^{2} x^{2} + 1} - \log \left (c^{2} x^{2} + 1\right )\right )}}{2 \, c^{2}} - \frac{{\left (3 \, c^{2} x^{2} - 2 \,{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 6 \, \sqrt{c^{2} x^{2} + 1} - 3 \, \log \left (c^{2} x^{2} + 1\right ) + 3\right )} b e^{2}}{24 \, c^{4}} - \frac{2 \, b c^{2} e^{2} x^{4} \log \left (c\right ) + 4 \, b c^{2} d^{2} \log \left (x\right )^{2} +{\left (8 \, c^{2} d e \log \left (c\right ) - e^{2}\right )} b x^{2} + 2 \,{\left (b c^{2} e^{2} x^{4} + 4 \, b c^{2} d e x^{2}\right )} \log \left (x\right ) - 2 \,{\left (b c^{2} e^{2} x^{4} + 4 \, b c^{2} d e x^{2} + 4 \, b c^{2} d^{2} \log \left (x\right )\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{8 \, c^{2}} + \frac{{\left (4 \, c^{2} d e - e^{2}\right )} b \log \left (c^{2} x^{2} + 1\right )}{8 \, c^{4}} \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^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\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 )^{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 (e x^{2} + d\right )}^{2}{\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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