Optimal. Leaf size=54 \[ -\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
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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 = {3771, 2639} \[ -\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx &=\frac {\int \sqrt {i \sinh (a+b x)} \, dx}{\sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ &=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 50, normalized size = 0.93 \[ \frac {2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)} E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (a+b x)\right )\right |2\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {\operatorname {csch}\left (b x + a\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}\left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 108, normalized size = 2.00 \[ \frac {\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b} \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}\left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}{\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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