Optimal. Leaf size=19 \[ -\frac{1}{f \sqrt{a \cosh ^2(e+f x)}} \]
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Rubi [A] time = 0.0696189, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3176, 3205, 16, 32} \[ -\frac{1}{f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 16
Rule 32
Rubi steps
\begin{align*} \int \frac{\tanh (e+f x)}{\sqrt{a+a \sinh ^2(e+f x)}} \, dx &=\int \frac{\tanh (e+f x)}{\sqrt{a \cosh ^2(e+f x)}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a x}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{(a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac{1}{f \sqrt{a \cosh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0317421, size = 19, normalized size = 1. \[ -\frac{1}{f \sqrt{a \cosh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 20, normalized size = 1.1 \begin{align*} -{\frac{1}{f}{\frac{1}{\sqrt{a+a \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77428, size = 45, normalized size = 2.37 \begin{align*} -\frac{2 \, e^{\left (-f x - e\right )}}{{\left (\sqrt{a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt{a}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75363, size = 428, normalized size = 22.53 \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{\tanh{\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 = 1.25999, size = 39, normalized size = 2.05 \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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