Optimal. Leaf size=80 \[ -\frac {2 \cosh (a+b x)}{3 b \sinh ^{\frac {3}{2}}(a+b x)}+\frac {2 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b \sqrt {\sinh (a+b x)}} \]
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Rubi [A] time = 0.03, 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 = {2636, 2642, 2641} \[ -\frac {2 \cosh (a+b x)}{3 b \sinh ^{\frac {3}{2}}(a+b x)}+\frac {2 i \sqrt {i \sinh (a+b x)} F\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{3 b \sqrt {\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{\frac {5}{2}}(a+b x)} \, dx &=-\frac {2 \cosh (a+b x)}{3 b \sinh ^{\frac {3}{2}}(a+b x)}-\frac {1}{3} \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx\\ &=-\frac {2 \cosh (a+b x)}{3 b \sinh ^{\frac {3}{2}}(a+b x)}-\frac {\sqrt {i \sinh (a+b x)} \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{3 \sqrt {\sinh (a+b x)}}\\ &=-\frac {2 \cosh (a+b x)}{3 b \sinh ^{\frac {3}{2}}(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)}}{3 b \sqrt {\sinh (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 86, normalized size = 1.08 \[ -\frac {2 \left (\sinh (a+b x) \sqrt {-\sinh (2 a+2 b x)-\cosh (2 a+2 b x)+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\cosh (2 (a+b x))+\sinh (2 (a+b x))\right )+\cosh (a+b x)\right )}{3 b \sinh ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sinh \left (b x + a\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {1}{\sinh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, 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 \[ \int \frac {1}{\sinh \left (b x + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {sinh}\left (a+b\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\sinh ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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