Optimal. Leaf size=29 \[ \frac{\sinh (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}} \]
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Rubi [A] time = 0.0442715, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3190, 191} \[ \frac{\sinh (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 191
Rubi steps
\begin{align*} \int \frac{\cosh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\sinh (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0279143, size = 29, normalized size = 1. \[ \frac{\sinh (e+f x)}{a f \sqrt{a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 28, normalized size = 1. \begin{align*}{\frac{\sinh \left ( fx+e \right ) }{af}{\frac{1}{\sqrt{a+b \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 [B] time = 1.65636, size = 319, normalized size = 11. \begin{align*} \frac{b^{2} e^{\left (-6 \, f x - 6 \, e\right )} + 2 \, a b - b^{2} +{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \,{\left (2 \, a b - b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{2 \,{\left (a^{2} - a b\right )}{\left (2 \,{\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac{3}{2}} f} - \frac{b^{2} + 3 \,{\left (2 \, a b - b^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )} +{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )} +{\left (2 \, a b - b^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{2 \,{\left (a^{2} - a b\right )}{\left (2 \,{\left (2 \, a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + b e^{\left (-4 \, f x - 4 \, e\right )} + b\right )}^{\frac{3}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82896, size = 641, normalized size = 22.1 \begin{align*} \frac{\sqrt{2}{\left (\cosh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2} - 1\right )} \sqrt{\frac{b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{a b f \cosh \left (f x + e\right )^{4} + 4 \, a b f \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + a b f \sinh \left (f x + e\right )^{4} + 2 \,{\left (2 \, a^{2} - a b\right )} f \cosh \left (f x + e\right )^{2} + a b f + 2 \,{\left (3 \, a b f \cosh \left (f x + e\right )^{2} +{\left (2 \, a^{2} - a b\right )} f\right )} \sinh \left (f x + e\right )^{2} + 4 \,{\left (a b f \cosh \left (f x + e\right )^{3} +{\left (2 \, a^{2} - a b\right )} 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(-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 [B] time = 1.2445, size = 211, normalized size = 7.28 \begin{align*} \frac{\frac{{\left (a^{3} f - 2 \, a^{2} b f + a b^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a^{4} b^{3} - 2 \, a^{3} b^{4} + a^{2} b^{5}} - \frac{a^{3} f - 2 \, a^{2} b f + a b^{2} f}{a^{4} b^{3} - 2 \, a^{3} b^{4} + a^{2} b^{5}}}{256 \, \sqrt{b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}} + \frac{f}{256 \, a b^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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