Optimal. Leaf size=25 \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}} \]
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Rubi [A] time = 0.104443, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3176, 3207, 2606, 8} \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{\coth ^2(e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx &=\int \frac{\coth ^2(e+f x)}{\sqrt{a \cosh ^2(e+f x)}} \, dx\\ &=\frac{\cosh (e+f x) \int \coth (e+f x) \text{csch}(e+f x) \, dx}{\sqrt{a \cosh ^2(e+f x)}}\\ &=-\frac{(i \cosh (e+f x)) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(e+f x))}{f \sqrt{a \cosh ^2(e+f x)}}\\ &=-\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0373885, size = 25, normalized size = 1. \[ -\frac{\coth (e+f x)}{f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 32, normalized size = 1.3 \begin{align*} -{\frac{\cosh \left ( fx+e \right ) }{\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 = 1.84918, size = 136, normalized size = 5.44 \begin{align*} \frac{\frac{\arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt{a}} + \frac{\sqrt{a} e^{\left (-f x - e\right )}}{a e^{\left (-2 \, f x - 2 \, e\right )} - a}}{f} - \frac{\arctan \left (e^{\left (-f x - e\right )}\right )}{\sqrt{a} f} + \frac{\sqrt{a} e^{\left (-f x - e\right )}}{{\left (a e^{\left (-2 \, f x - 2 \, e\right )} - a\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87446, size = 428, normalized size = 17.12 \begin{align*} -\frac{2 \, \sqrt{a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a}{\left (\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 )} e^{\left (-f x - e\right )}}{a f \cosh \left (f x + e\right )^{2} +{\left (a f e^{\left (2 \, f x + 2 \, e\right )} + a f\right )} \sinh \left (f x + e\right )^{2} - a f +{\left (a f \cosh \left (f x + e\right )^{2} - a f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (a f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a 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 )}}{\sqrt{a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37947, size = 39, normalized size = 1.56 \begin{align*} -\frac{2 \, e^{\left (f x + e\right )}}{\sqrt{a} f{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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