Optimal. Leaf size=190 \[ -\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )} \]
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Rubi [A] time = 0.29, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )} \]
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 )^2} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^3 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)^2}+\frac {\sinh (3 x)}{4 (a+b x)^2}\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)^2} \, 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)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}-\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {\left (c^3 \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 {sech}^{-1}(c x)\right )}{4 b}-\frac {\left (3 c^3 \cosh \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 )}{4 b}+\frac {\left (c^3 \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 {sech}^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3 \sinh \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 )}{4 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 250, normalized size = 1.32 \[ \frac {-c^3 x^3 \cosh \left (\frac {a}{b}\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )-3 c^3 x^3 \cosh \left (\frac {3 a}{b}\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+a c^3 x^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+b c^3 x^3 \sinh \left (\frac {a}{b}\right ) \text {sech}^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+3 a c^3 x^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+3 b c^3 x^3 \sinh \left (\frac {3 a}{b}\right ) \text {sech}^{-1}(c x) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+4 b c x \sqrt {\frac {1-c x}{c x+1}}+4 b \sqrt {\frac {1-c x}{c x+1}}}{4 b^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x^{4} \operatorname {arsech}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arsech}\left (c x\right ) + a^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 420, normalized size = 2.21 \[ c^{3} \left (-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -3 c^{2} x^{2}+4}{8 c^{3} x^{3} b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1}{8 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{8 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4}{8 b \,c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}\right ) \]
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 \[ -\frac {c^{2} x^{3} + {\left (c^{2} x^{3} - x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - x}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{4} \log \relax (x) + {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} - b^{2} \log \relax (c) + a b\right )} x^{4} - {\left (b^{2} x^{4} \log \relax (x) + {\left (b^{2} \log \relax (c) - a b\right )} x^{4}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + {\left (\sqrt {c x + 1} \sqrt {-c x + 1} b^{2} x^{4} - {\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{4}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )} - \int \frac {3 \, c^{4} x^{4} - 6 \, c^{2} x^{2} + {\left (c^{2} x^{2} - 3\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + {\left (2 \, c^{4} x^{4} - 7 \, c^{2} x^{2} + 6\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + 3}{{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{4} \log \relax (x) + {\left ({\left (b^{2} c^{4} \log \relax (c) - a b c^{4}\right )} x^{4} - 2 \, {\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - a b\right )} x^{4} - {\left (b^{2} x^{4} \log \relax (x) + {\left (b^{2} \log \relax (c) - a b\right )} x^{4}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - 2 \, {\left ({\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{4} \log \relax (x) + {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} - b^{2} \log \relax (c) + a b\right )} x^{4}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + {\left ({\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} x^{4} + 2 \, {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{4} - {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{4}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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