Optimal. Leaf size=194 \[ -\frac{b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c^3}+\frac{b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,-e^{\text{csch}^{-1}(c x)}\right )}{c^3}-\frac{b^3 \text{PolyLog}\left (3,e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}-\frac{b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2}{c^3}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{b^3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c^3} \]
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Rubi [A] time = 0.208091, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6286, 5452, 4186, 3770, 4182, 2531, 2282, 6589} \[ -\frac{b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c^3}+\frac{b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,-e^{\text{csch}^{-1}(c x)}\right )}{c^3}-\frac{b^3 \text{PolyLog}\left (3,e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}-\frac{b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2}{c^3}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{b^3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 6286
Rule 5452
Rule 4186
Rule 3770
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \left (a+b \text{csch}^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \coth (x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}\\ &=\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3+\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{2 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{b^2 \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}+\frac{b^2 \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}-\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{c^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{b \left (a+b \text{csch}^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}+\frac{b^3 \text{Li}_3\left (-e^{\text{csch}^{-1}(c x)}\right )}{c^3}-\frac{b^3 \text{Li}_3\left (e^{\text{csch}^{-1}(c x)}\right )}{c^3}\\ \end{align*}
Mathematica [B] time = 7.48386, size = 548, normalized size = 2.82 \[ \frac{a b^2 \left (2 c^3 x^3 \left (-\frac{4 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c x)}\right )}{c^3 x^3}+4 \text{csch}^{-1}(c x)^2+2 \cosh \left (2 \text{csch}^{-1}(c x)\right )-\frac{3 \text{csch}^{-1}(c x) \log \left (1-e^{-\text{csch}^{-1}(c x)}\right )}{c x}+\frac{3 \text{csch}^{-1}(c x) \log \left (e^{-\text{csch}^{-1}(c x)}+1\right )}{c x}+2 \text{csch}^{-1}(c x) \sinh \left (2 \text{csch}^{-1}(c x)\right )+\text{csch}^{-1}(c x) \log \left (1-e^{-\text{csch}^{-1}(c x)}\right ) \sinh \left (3 \text{csch}^{-1}(c x)\right )-\text{csch}^{-1}(c x) \log \left (e^{-\text{csch}^{-1}(c x)}+1\right ) \sinh \left (3 \text{csch}^{-1}(c x)\right )-2\right )+8 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c x)}\right )\right )}{8 c^3}+\frac{b^3 \left (48 \text{csch}^{-1}(c x) \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c x)}\right )-48 \text{csch}^{-1}(c x) \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c x)}\right )+48 \text{PolyLog}\left (3,-e^{-\text{csch}^{-1}(c x)}\right )-48 \text{PolyLog}\left (3,e^{-\text{csch}^{-1}(c x)}\right )+16 c^3 x^3 \text{csch}^{-1}(c x)^3 \sinh ^4\left (\frac{1}{2} \text{csch}^{-1}(c x)\right )+\frac{\text{csch}^{-1}(c x)^3 \text{csch}^4\left (\frac{1}{2} \text{csch}^{-1}(c x)\right )}{c x}+6 \text{csch}^{-1}(c x)^2 \text{csch}^2\left (\frac{1}{2} \text{csch}^{-1}(c x)\right )-4 \text{csch}^{-1}(c x)^3 \coth \left (\frac{1}{2} \text{csch}^{-1}(c x)\right )+24 \text{csch}^{-1}(c x) \coth \left (\frac{1}{2} \text{csch}^{-1}(c x)\right )+24 \text{csch}^{-1}(c x)^2 \log \left (1-e^{-\text{csch}^{-1}(c x)}\right )-24 \text{csch}^{-1}(c x)^2 \log \left (e^{-\text{csch}^{-1}(c x)}+1\right )+4 \text{csch}^{-1}(c x)^3 \tanh \left (\frac{1}{2} \text{csch}^{-1}(c x)\right )-24 \text{csch}^{-1}(c x) \tanh \left (\frac{1}{2} \text{csch}^{-1}(c x)\right )+6 \text{csch}^{-1}(c x)^2 \text{sech}^2\left (\frac{1}{2} \text{csch}^{-1}(c x)\right )-48 \log \left (\tanh \left (\frac{1}{2} \text{csch}^{-1}(c x)\right )\right )\right )}{48 c^3}+\frac{a^2 b x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{2 c}-\frac{a^2 b \log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{2 c^3}+a^2 b x^3 \text{csch}^{-1}(c x)+\frac{a^3 x^3}{3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{3}\, 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*} \text{result too large to display} \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 (b^{3} x^{2} \operatorname{arcsch}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname{arcsch}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname{arcsch}\left (c x\right ) + a^{3} x^{2}, 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 x^{2} \left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{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{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{3} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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