Optimal. Leaf size=54 \[ -\frac {2 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{b \sqrt {\sinh (a+b x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2721, 2720}
\begin {gather*} -\frac {2 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sinh (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx &=\frac {\sqrt {i \sinh (a+b x)} \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{\sqrt {\sinh (a+b x)}}\\ &=-\frac {2 i F\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 48, normalized size = 0.89 \begin {gather*} -\frac {2 F\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {i \sinh (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 87, normalized size = 1.61
method | result | size |
default | \(\frac {i \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(87\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 24, normalized size = 0.44 \begin {gather*} \frac {2 \, \sqrt {2} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \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}{\sqrt {\sinh {\left (a + b x \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}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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