Optimal. Leaf size=87 \[ \frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}+\frac {10 i \sqrt {i \sinh (x)} \sinh (x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3207, 2636, 2642, 2641} \[ \frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}+\frac {10 i \sqrt {i \sinh (x)} \sinh (x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx &=\frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}-\frac {\left (5 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {5}{2}}(x)} \, dx}{7 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {\left (5 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sinh (x)}} \, dx}{21 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {\left (5 \sqrt {i \sinh (x)} \sinh (x)\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx}{21 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {10 i F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)} \sinh (x)}{21 a \sqrt {a \sinh ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 53, normalized size = 0.61 \[ \frac {2 \left (5 \cosh (x)-3 \coth (x) \text {csch}(x)+5 (i \sinh (x))^{3/2} F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right )}{21 a \sqrt {a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sinh \relax (x)^{3}}}{a^{2} \sinh \relax (x)^{6}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \sinh \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \left (\sinh ^{3}\relax (x )\right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (a \sinh \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,{\mathrm {sinh}\relax (x)}^3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sinh ^{3}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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