Optimal. Leaf size=64 \[ -\frac{\coth (e+f x)}{a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{a f \sqrt{a \cosh ^2(e+f x)}} \]
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Rubi [A] time = 0.131732, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3176, 3207, 2621, 321, 207} \[ -\frac{\coth (e+f x)}{a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\cosh (e+f x) \tan ^{-1}(\sinh (e+f x))}{a f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2621
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \frac{\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\coth ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac{\cosh (e+f x) \int \text{csch}^2(e+f x) \text{sech}(e+f x) \, dx}{a \sqrt{a \cosh ^2(e+f x)}}\\ &=-\frac{(i \cosh (e+f x)) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,-i \text{csch}(e+f x)\right )}{a f \sqrt{a \cosh ^2(e+f x)}}\\ &=-\frac{\coth (e+f x)}{a f \sqrt{a \cosh ^2(e+f x)}}-\frac{(i \cosh (e+f x)) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,-i \text{csch}(e+f x)\right )}{a f \sqrt{a \cosh ^2(e+f x)}}\\ &=-\frac{\tan ^{-1}(\sinh (e+f x)) \cosh (e+f x)}{a f \sqrt{a \cosh ^2(e+f x)}}-\frac{\coth (e+f x)}{a f \sqrt{a \cosh ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0787, size = 46, normalized size = 0.72 \[ -\frac{\coth (e+f x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\sinh ^2(e+f x)\right )}{a f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 51, normalized size = 0.8 \begin{align*} -{\frac{\cosh \left ( fx+e \right ) \left ( \arctan \left ( \sinh \left ( fx+e \right ) \right ) \sinh \left ( fx+e \right ) +1 \right ) }{a\sinh \left ( fx+e \right ) f}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.71757, size = 433, normalized size = 6.77 \begin{align*} -\frac{\frac{3 \, \sqrt{a} e^{\left (-f x - e\right )} + 2 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}}{a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}} - \frac{3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac{3}{2}}}}{2 \, f} - \frac{5 \, \sqrt{a} e^{\left (-f x - e\right )} + 6 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} - 3 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}}{4 \,{\left (a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}\right )} f} + \frac{3 \, \sqrt{a} e^{\left (-f x - e\right )} - 6 \, \sqrt{a} e^{\left (-3 \, f x - 3 \, e\right )} - 5 \, \sqrt{a} e^{\left (-5 \, f x - 5 \, e\right )}}{4 \,{\left (a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}\right )} f} + \frac{\arctan \left (e^{\left (-f x - e\right )}\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 = 1.80605, size = 652, normalized size = 10.19 \begin{align*} -\frac{2 \,{\left ({\left (2 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} +{\left (\cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )}\right )} \arctan \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + \cosh \left (f x + e\right ) e^{\left (f x + e\right )} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )\right )} \sqrt{a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f +{\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{2} +{\left (a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (a^{2} f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\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{\coth ^{2}{\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 [A] time = 1.42971, size = 58, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (\frac{\arctan \left (e^{\left (f x + e\right )}\right )}{a^{\frac{3}{2}}} + \frac{e^{\left (f x + e\right )}}{a^{\frac{3}{2}}{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}}\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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