Optimal. Leaf size=62 \[ \frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \sqrt {i \sinh (x)} \text {csch}^2(x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{3 \sqrt {a \text {csch}^3(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3769, 3771, 2641} \[ \frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \sqrt {i \sinh (x)} \text {csch}^2(x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{3 \sqrt {a \text {csch}^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx &=\frac {(i \text {csch}(x))^{3/2} \int \frac {1}{(i \text {csch}(x))^{3/2}} \, dx}{\sqrt {a \text {csch}^3(x)}}\\ &=\frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}+\frac {(i \text {csch}(x))^{3/2} \int \sqrt {i \text {csch}(x)} \, dx}{3 \sqrt {a \text {csch}^3(x)}}\\ &=\frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {\left (\text {csch}^2(x) \sqrt {i \sinh (x)}\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx}{3 \sqrt {a \text {csch}^3(x)}}\\ &=\frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \text {csch}^2(x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)}}{3 \sqrt {a \text {csch}^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 43, normalized size = 0.69 \[ \frac {2 \left (\coth (x)+\frac {\text {csch}(x) F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )}{\sqrt {i \sinh (x)}}\right )}{3 \sqrt {a \text {csch}^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \operatorname {csch}\relax (x)^{3}}}{a \operatorname {csch}\relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \operatorname {csch}\relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \mathrm {csch}\relax (x )^{3}}}\, dx \]
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 \[ \int \frac {1}{\sqrt {a \operatorname {csch}\relax (x)^{3}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {a}{{\mathrm {sinh}\relax (x)}^3}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \operatorname {csch}^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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