Optimal. Leaf size=66 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 a^{3/2} f}-\frac{\text{csch}^2(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 a^2 f} \]
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Rubi [A] time = 0.139467, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3176, 3205, 16, 51, 63, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 a^{3/2} f}-\frac{\text{csch}^2(e+f x) \sqrt{a \cosh ^2(e+f x)}}{2 a^2 f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\coth ^3(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\coth ^3(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(1-x)^2 (a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x)^2 \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 a f}\\ &=-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^2(e+f x)}{2 a^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 a f}\\ &=-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^2(e+f x)}{2 a^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a \cosh ^2(e+f x)}\right )}{2 a^2 f}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 a^{3/2} f}-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^2(e+f x)}{2 a^2 f}\\ \end{align*}
Mathematica [A] time = 0.131715, size = 67, normalized size = 1.02 \[ -\frac{\cosh ^3(e+f x) \left (\text{csch}^2\left (\frac{1}{2} (e+f x)\right )+\text{sech}^2\left (\frac{1}{2} (e+f x)\right )+4 \log \left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )\right )}{8 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.069, size = 36, normalized size = 0.6 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{1}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{3}a}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.11193, size = 135, normalized size = 2.05 \begin{align*} \frac{e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}}{{\left (2 \, a^{\frac{3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} - a^{\frac{3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} - a^{\frac{3}{2}}\right )} f} + \frac{\log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, a^{\frac{3}{2}} f} - \frac{\log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, a^{\frac{3}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87461, size = 1458, normalized size = 22.09 \begin{align*} \text{result too large to display} \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{\coth ^{3}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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