Optimal. Leaf size=21 \[ -\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.0774936, antiderivative size = 21, 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} \[ -\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
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)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\tanh (e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{(a x)^{5/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0388074, size = 21, normalized size = 1. \[ -\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 20, normalized size = 1. \begin{align*} -{\frac{1}{3\,f} \left ( a+a \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.8976, size = 82, normalized size = 3.9 \begin{align*} -\frac{8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \,{\left (3 \, a^{\frac{3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, a^{\frac{3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + a^{\frac{3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac{3}{2}}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84687, size = 1511, normalized size = 71.95 \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{\tanh{\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 = 1.31568, size = 43, normalized size = 2.05 \begin{align*} -\frac{8 \, e^{\left (3 \, f x + 3 \, e\right )}}{3 \, a^{\frac{3}{2}} f{\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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