Optimal. Leaf size=118 \[ -\frac{10 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right ) \sqrt{b \text{csch}(c+d x)}}{21 b^4 d}-\frac{10 \cosh (c+d x)}{21 b^3 d \sqrt{b \text{csch}(c+d x)}}+\frac{2 \cosh (c+d x)}{7 b d (b \text{csch}(c+d x))^{5/2}} \]
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Rubi [A] time = 0.0564636, antiderivative size = 118, 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 = {3769, 3771, 2641} \[ -\frac{10 \cosh (c+d x)}{21 b^3 d \sqrt{b \text{csch}(c+d x)}}-\frac{10 i \sqrt{i \sinh (c+d x)} F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \text{csch}(c+d x)}}{21 b^4 d}+\frac{2 \cosh (c+d x)}{7 b d (b \text{csch}(c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(b \text{csch}(c+d x))^{7/2}} \, dx &=\frac{2 \cosh (c+d x)}{7 b d (b \text{csch}(c+d x))^{5/2}}-\frac{5 \int \frac{1}{(b \text{csch}(c+d x))^{3/2}} \, dx}{7 b^2}\\ &=\frac{2 \cosh (c+d x)}{7 b d (b \text{csch}(c+d x))^{5/2}}-\frac{10 \cosh (c+d x)}{21 b^3 d \sqrt{b \text{csch}(c+d x)}}+\frac{5 \int \sqrt{b \text{csch}(c+d x)} \, dx}{21 b^4}\\ &=\frac{2 \cosh (c+d x)}{7 b d (b \text{csch}(c+d x))^{5/2}}-\frac{10 \cosh (c+d x)}{21 b^3 d \sqrt{b \text{csch}(c+d x)}}+\frac{\left (5 \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}\right ) \int \frac{1}{\sqrt{i \sinh (c+d x)}} \, dx}{21 b^4}\\ &=\frac{2 \cosh (c+d x)}{7 b d (b \text{csch}(c+d x))^{5/2}}-\frac{10 \cosh (c+d x)}{21 b^3 d \sqrt{b \text{csch}(c+d x)}}-\frac{10 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)}}{21 b^4 d}\\ \end{align*}
Mathematica [A] time = 0.156832, size = 80, normalized size = 0.68 \[ \frac{\sqrt{b \text{csch}(c+d x)} \left (40 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{4} (-2 i c-2 i d x+\pi ),2\right )-26 \sinh (2 (c+d x))+3 \sinh (4 (c+d x))\right )}{84 b^4 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.073, 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 \frac{1}{\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 (\frac{\sqrt{b \operatorname{csch}\left (d x + c\right )}}{b^{4} \operatorname{csch}\left (d x + c\right )^{4}}, 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 \frac{1}{\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 \frac{1}{\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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