Optimal. Leaf size=80 \[ -\frac {2 \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 i \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3768, 3771, 2641} \[ -\frac {2 \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 i \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \text {csch}^{\frac {5}{2}}(a+b x) \, dx &=-\frac {2 \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}-\frac {1}{3} \int \sqrt {\text {csch}(a+b x)} \, dx\\ &=-\frac {2 \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}-\frac {1}{3} \left (\sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx\\ &=-\frac {2 \cosh (a+b x) \text {csch}^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 i \sqrt {\text {csch}(a+b x)} 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)}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 61, normalized size = 0.76 \[ -\frac {2 \sqrt {\text {csch}(a+b x)} \left (\coth (a+b x)+i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}, x\right ) \]
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 \[ \int \operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 101, normalized size = 1.26 \[ -\frac {i \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (b x +a \right )+2 \left (\cosh ^{2}\left (b x +a \right )\right )}{3 \sinh \left (b x +a \right )^{\frac {3}{2}} \cosh \left (b x +a \right ) b} \]
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 \[ \int \operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{5/2} \,d x \]
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 \[ \int \operatorname {csch}^{\frac {5}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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