Optimal. Leaf size=87 \[ \frac{3 \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 f}-\frac{\text{csch}^2(e+f x) \left (a \cosh ^2(e+f x)\right )^{3/2}}{2 a f} \]
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Rubi [A] time = 0.142148, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3176, 3205, 16, 47, 50, 63, 206} \[ \frac{3 \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 f}-\frac{\text{csch}^2(e+f x) \left (a \cosh ^2(e+f x)\right )^{3/2}}{2 a f} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 47
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \coth ^3(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt{a \cosh ^2(e+f x)} \coth ^3(e+f x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \sqrt{a x}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a x)^{3/2}}{(1-x)^2} \, dx,x,\cosh ^2(e+f x)\right )}{2 a f}\\ &=-\frac{\left (a \cosh ^2(e+f x)\right )^{3/2} \text{csch}^2(e+f x)}{2 a f}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{a x}}{1-x} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=\frac{3 \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{\left (a \cosh ^2(e+f x)\right )^{3/2} \text{csch}^2(e+f x)}{2 a f}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{4 f}\\ &=\frac{3 \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{\left (a \cosh ^2(e+f x)\right )^{3/2} \text{csch}^2(e+f x)}{2 a f}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a \cosh ^2(e+f x)}\right )}{2 f}\\ &=-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \cosh ^2(e+f x)}}{\sqrt{a}}\right )}{2 f}+\frac{3 \sqrt{a \cosh ^2(e+f x)}}{2 f}-\frac{\left (a \cosh ^2(e+f x)\right )^{3/2} \text{csch}^2(e+f x)}{2 a f}\\ \end{align*}
Mathematica [A] time = 0.260867, size = 77, normalized size = 0.89 \[ \frac{\text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)} \left (8 \cosh (e+f x)-\text{csch}^2\left (\frac{1}{2} (e+f x)\right )-\text{sech}^2\left (\frac{1}{2} (e+f x)\right )+12 \log \left (\tanh \left (\frac{1}{2} (e+f x)\right )\right )\right )}{8 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.092, size = 54, normalized size = 0.6 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{a \left ( \cosh \left ( fx+e \right ) \right ) ^{4}}{\sinh \left ( fx+e \right ) \left ( \left ( \cosh \left ( fx+e \right ) \right ) ^{2}-1 \right ) }{\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 = 1.94735, size = 170, normalized size = 1.95 \begin{align*} -\frac{3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, f} + \frac{3 \, \sqrt{a} \log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, f} - \frac{3 \, \sqrt{a} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, \sqrt{a} e^{\left (-4 \, f x - 4 \, e\right )} - \sqrt{a} e^{\left (-6 \, f x - 6 \, e\right )} - \sqrt{a}}{2 \, f{\left (e^{\left (-f x - e\right )} - 2 \, e^{\left (-3 \, f x - 3 \, e\right )} + e^{\left (-5 \, f x - 5 \, e\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97631, size = 2056, normalized size = 23.63 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25896, size = 120, normalized size = 1.38 \begin{align*} -\frac{\sqrt{a}{\left (\frac{2 \,{\left (e^{\left (3 \, f x + 3 \, e\right )} + e^{\left (f x + e\right )}\right )}}{{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{2}} - e^{\left (f x + e\right )} - e^{\left (-f x - e\right )} + 3 \, \log \left (e^{\left (f x + e\right )} + 1\right ) - 3 \, \log \left ({\left | e^{\left (f x + e\right )} - 1 \right |}\right )\right )}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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