Optimal. Leaf size=116 \[ \frac{6 b^3 \cosh (c+d x) \sqrt{b \text{csch}(c+d x)}}{5 d}+\frac{6 i b^4 E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}}-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.0611193, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2639} \[ \frac{6 b^3 \cosh (c+d x) \sqrt{b \text{csch}(c+d x)}}{5 d}+\frac{6 i b^4 E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}}-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int (b \text{csch}(c+d x))^{7/2} \, dx &=-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{5/2}}{5 d}-\frac{1}{5} \left (3 b^2\right ) \int (b \text{csch}(c+d x))^{3/2} \, dx\\ &=\frac{6 b^3 \cosh (c+d x) \sqrt{b \text{csch}(c+d x)}}{5 d}-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{5/2}}{5 d}-\frac{1}{5} \left (3 b^4\right ) \int \frac{1}{\sqrt{b \text{csch}(c+d x)}} \, dx\\ &=\frac{6 b^3 \cosh (c+d x) \sqrt{b \text{csch}(c+d x)}}{5 d}-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{5/2}}{5 d}-\frac{\left (3 b^4\right ) \int \sqrt{i \sinh (c+d x)} \, dx}{5 \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}}\\ &=\frac{6 b^3 \cosh (c+d x) \sqrt{b \text{csch}(c+d x)}}{5 d}-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{5/2}}{5 d}+\frac{6 i b^4 E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{5 d \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.157263, size = 79, normalized size = 0.68 \[ -\frac{2 b^3 \sqrt{b \text{csch}(c+d x)} \left (-3 \cosh (c+d x)+\coth (c+d x) \text{csch}(c+d x)+3 \sqrt{i \sinh (c+d x)} E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{5 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm csch} \left (dx+c\right ) \right ) ^{{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{csch}\left (d x + c\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \operatorname{csch}\left (d x + c\right )} b^{3} \operatorname{csch}\left (d x + c\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{csch}{\left (c + d x \right )}\right )^{\frac{7}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{csch}\left (d x + c\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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