Optimal. Leaf size=112 \[ \frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6420, 5556, 12,
3378, 3384, 3379, 3382} \begin {gather*} \frac {c^2 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3}+\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 6420
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx &=-\left (c^2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (\frac {1}{2} c^2 \text {Subst}\left (\int \frac {\sinh (2 x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \text {Subst}\left (\int \frac {\cosh (2 x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 b}\\ &=\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b^2}\\ &=\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {\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 {sech}^{-1}(c x)\right )}{b^2}+\frac {\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 {sech}^{-1}(c x)\right )}{b^2}\\ &=\frac {c^2 \cosh \left (2 \text {sech}^{-1}(c x)\right )}{2 b^2 \left (a+b \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 )}{b^3}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 122, normalized size = 1.09 \begin {gather*} \frac {\frac {b^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {b \left (2-c^2 x^2\right )}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )}+2 c^2 \left (\text {Chi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs.
\(2(108)=216\).
time = 0.43, size = 277, normalized size = 2.47
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {\left (2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +c^{2} x^{2}-2\right ) \left (2 b \,\mathrm {arcsech}\left (c x \right )+2 a -b \right )}{8 c^{2} x^{2} b^{2} \left (b^{2} \mathrm {arcsech}\left (c x \right )^{2}+2 a b \,\mathrm {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, \frac {2 a}{b}+2 \,\mathrm {arcsech}\left (c x \right )\right )}{2 b^{3}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{8 b \,c^{2} x^{2} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{4 b^{2} c^{2} x^{2} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{3}}\right )\) | \(277\) |
default | \(c^{2} \left (-\frac {\left (2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +c^{2} x^{2}-2\right ) \left (2 b \,\mathrm {arcsech}\left (c x \right )+2 a -b \right )}{8 c^{2} x^{2} b^{2} \left (b^{2} \mathrm {arcsech}\left (c x \right )^{2}+2 a b \,\mathrm {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, \frac {2 a}{b}+2 \,\mathrm {arcsech}\left (c x \right )\right )}{2 b^{3}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{8 b \,c^{2} x^{2} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x}{4 b^{2} c^{2} x^{2} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{3}}\right )\) | \(277\) |
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 {asech}{\left (c x \right )}\right )^{3}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{x^3\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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