Optimal. Leaf size=117 \[ \frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}+\frac {c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b} \]
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Rubi [A] time = 0.24, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6285, 5448, 3303, 3298, 3301} \[ \frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}+\frac {c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 6285
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^3 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)}+\frac {\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (\frac {1}{4} c^3 \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\right )-\frac {1}{4} c^3 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\left (\frac {1}{4} \left (c^3 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\right )-\frac {1}{4} \left (c^3 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{4} \left (c^3 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{4} \left (c^3 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {c^3 \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b}+\frac {c^3 \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b}-\frac {c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 91, normalized size = 0.78 \[ -\frac {c^3 \left (\sinh \left (\frac {a}{b}\right ) \left (-\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{4} \operatorname {arsech}\left (c x\right ) + a x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 110, normalized size = 0.94 \[ c^{3} \left (-\frac {{\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{8 b}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{8 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b}+\frac {{\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b}\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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