Optimal. Leaf size=63 \[ \frac{c^2 \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )}{2 b}-\frac{c^2 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )}{2 b} \]
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Rubi [A] time = 0.140565, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6286, 5448, 12, 3303, 3298, 3301} \[ \frac{c^2 \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )}{2 b}-\frac{c^2 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 6286
Rule 5448
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \text{csch}^{-1}(c x)\right )} \, dx &=-\left (c^2 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\left (c^2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 (a+b x)} \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\left (\frac{1}{2} c^2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\left (\frac{1}{2} \left (c^2 \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\right )+\frac{1}{2} \left (c^2 \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{c^2 \text{Chi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right ) \sinh \left (\frac{2 a}{b}\right )}{2 b}-\frac{c^2 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0715245, size = 56, normalized size = 0.89 \[ \frac{c^2 \left (\sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )-\cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) }}\, 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*} \int \frac{1}{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{3}}\,{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{1}{b x^{3} \operatorname{arcsch}\left (c x\right ) + a x^{3}}, 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{1}{x^{3} \left (a + b \operatorname{acsch}{\left (c x \right )}\right )}\, 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{1}{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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