Optimal. Leaf size=63 \[ \frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right )}{2 b} \]
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Rubi [A]
time = 0.11, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 = {6421, 5556, 12,
3384, 3379, 3382} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 6421
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b \text {csch}^{-1}(c x)\right )} \, dx &=-\left (c^2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\left (c^2 \text {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 \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.05, size = 56, normalized size = 0.89 \begin {gather*} \frac {c^2 \left (\text {Chi}\left (\frac {2 a}{b}+2 \text {csch}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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