Optimal. Leaf size=90 \[ \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}+\frac {2 i \sqrt {i \sinh (c+d x)} F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \text {csch}(c+d x)}}{3 b^2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3769, 3771, 2641} \[ \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}+\frac {2 i \sqrt {i \sinh (c+d x)} F\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \text {csch}(c+d x)}}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx &=\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\int \sqrt {b \text {csch}(c+d x)} \, dx}{3 b^2}\\ &=\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\left (\sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}\right ) \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx}{3 b^2}\\ &=\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}+\frac {2 i \sqrt {b \text {csch}(c+d x)} F\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {i \sinh (c+d x)}}{3 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 73, normalized size = 0.81 \[ \frac {\text {csch}^2(c+d x) \left (\sinh (2 (c+d x))-2 i \sqrt {i \sinh (c+d x)} F\left (\left .\frac {1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{3 d (b \text {csch}(c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {csch}\left (d x + c\right )}}{b^{2} \operatorname {csch}\left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,\mathrm {csch}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \operatorname {csch}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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