Optimal. Leaf size=46 \[ \frac{c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )}{b} \]
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Rubi [A] time = 0.104262, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6286, 3303, 3298, 3301} \[ \frac{c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 6286
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \text{csch}^{-1}(c x)\right )} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\left (\left (c \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\right )+\left (c \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )}{b}+\frac{c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0723812, size = 44, normalized size = 0.96 \[ -\frac{c \left (\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{csch}^{-1}(c x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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^{2}}\,{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^{2} \operatorname{arcsch}\left (c x\right ) + a 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 \frac{1}{x^{2} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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