Optimal. Leaf size=63 \[ \frac {c^2 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{2 b}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{2 b} \]
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Rubi [A] time = 0.14, 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 = {6285, 5448, 12, 3303, 3298, 3301} \[ \frac {c^2 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{2 b}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 6285
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx &=-\left (c^2 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^2 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\text {sech}^{-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 {sech}^{-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 {sech}^{-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 {sech}^{-1}(c x)\right )\\ &=\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-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 {sech}^{-1}(c x)\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 56, normalized size = 0.89 \[ \frac {c^2 \left (\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{3} \operatorname {arsech}\left (c x\right ) + a x^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 60, normalized size = 0.95 \[ c^{2} \left (-\frac {{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, \frac {2 a}{b}+2 \,\mathrm {arcsech}\left (c x \right )\right )}{4 b}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \,\mathrm {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{4 b}\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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