Optimal. Leaf size=46 \[ \frac {c \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b} \]
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Rubi [A]
time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6420, 3384,
3379, 3382} \begin {gather*} \frac {c \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 6420
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )} \, dx &=-\left (c \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (\left (c \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\right )+\left (c \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {c \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 43, normalized size = 0.93 \begin {gather*} \frac {c \left (\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 54, normalized size = 1.17
method | result | size |
derivativedivides | \(c \left (-\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b}\right )\) | \(54\) |
default | \(c \left (-\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{2 b}+\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b}\right )\) | \(54\) |
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^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}\, 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 = -0.02 \begin {gather*} \int \frac {1}{x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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