Optimal. Leaf size=54 \[ -\frac {2 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sinh (a+b x)}} \]
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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 = {2642, 2641} \[ -\frac {2 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
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 = 0.12, size = 48, normalized size = 0.89 \[ -\frac {2 \sqrt {\sinh (a+b x)} F\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )}{b \sqrt {i \sinh (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {\sinh \left (b x + a\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {\sinh \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 87, normalized size = 1.61 \[ \frac {i \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b} \]
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 \[ \int \frac {1}{\sqrt {\sinh \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\sinh {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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