Optimal. Leaf size=240 \[ \frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}+\frac {9 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2} \]
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Rubi [A] time = 0.37, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6285, 5448, 3297, 3303, 3298, 3301} \[ \frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}+\frac {9 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 3297
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 )^3} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^3 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)^3}+\frac {\sinh (3 x)}{4 (a+b x)^3}\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)^3} \, 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)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \operatorname {Subst}\left (\int \frac {\cosh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b}-\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2}-\frac {\left (9 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {\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 )}{8 b^2}-\frac {\left (9 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 )}{8 b^2}+\frac {\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 )}{8 b^2}+\frac {\left (9 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 )}{8 b^2}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \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 )}{8 b^3}+\frac {9 c^3 \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 204, normalized size = 0.85 \[ \frac {\frac {4 b^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2}-8 c^3 \left (\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+9 c^3 \left (\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+\sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )+\frac {4 b \left (3-2 c^2 x^2\right )}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )}}{8 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{3} x^{4} \operatorname {arsech}\left (c x\right )^{3} + 3 \, a b^{2} x^{4} \operatorname {arsech}\left (c x\right )^{2} + 3 \, a^{2} b x^{4} \operatorname {arsech}\left (c x\right ) + a^{3} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 628, normalized size = 2.62 \[ c^{3} \left (\frac {\left (\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -3 c^{2} x^{2}+4\right ) \left (3 b \,\mathrm {arcsech}\left (c x \right )+3 a -b \right )}{16 c^{3} x^{3} b^{2} \left (\mathrm {arcsech}\left (c x \right )^{2} b^{2}+2 \,\mathrm {arcsech}\left (c x \right ) a b +a^{2}\right )}-\frac {9 \,{\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{16 b^{3}}-\frac {\left (\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1\right ) \left (b \,\mathrm {arcsech}\left (c x \right )+a -b \right )}{16 c x \,b^{2} \left (\mathrm {arcsech}\left (c x \right )^{2} b^{2}+2 \,\mathrm {arcsech}\left (c x \right ) a b +a^{2}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{16 b^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{16 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{16 b^{2} c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4}{16 b \,c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{2}}-\frac {3 \left (\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4\right )}{16 b^{2} c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {9 \,{\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{16 b^{3}}\right ) \]
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 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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