Optimal. Leaf size=65 \[ \frac{2 \sinh (e+f x)}{3 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.0579766, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3190, 192, 191} \[ \frac{2 \sinh (e+f x)}{3 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{\cosh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=\frac{\sinh (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{2 \sinh (e+f x)}{3 a^2 f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0449284, size = 47, normalized size = 0.72 \[ \frac{\sinh (e+f x) \left (3 a+2 b \sinh ^2(e+f x)\right )}{3 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{f} \left ({\frac{\sinh \left ( fx+e \right ) }{3\,a} \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\sinh \left ( fx+e \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6551, size = 655, normalized size = 10.08 \begin{align*} \frac{2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 5 \,{\left (4 \, a^{3} b - 6 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 2 \,{\left (24 \, a^{4} - 48 \, a^{3} b + 49 \, a^{2} b^{2} - 25 \, a b^{3} + 5 \, b^{4}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \,{\left (6 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right )} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \,{\left (4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-8 \, f x - 8 \, e\right )} +{\left (2 \, a b^{3} - b^{4}\right )} e^{\left (-10 \, f x - 10 \, e\right )}}{3 \,{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\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{5}{2}} f} - \frac{2 \, a b^{3} - b^{4} + 5 \,{\left (4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \,{\left (6 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right )} e^{\left (-4 \, f x - 4 \, e\right )} + 2 \,{\left (24 \, a^{4} - 48 \, a^{3} b + 49 \, a^{2} b^{2} - 25 \, a b^{3} + 5 \, b^{4}\right )} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \,{\left (4 \, a^{3} b - 6 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right )} e^{\left (-8 \, f x - 8 \, e\right )} +{\left (2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} e^{\left (-10 \, f x - 10 \, e\right )}}{3 \,{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\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{5}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.09934, size = 2161, normalized size = 33.25 \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(-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 = 1.69479, size = 702, normalized size = 10.8 \begin{align*} \frac{2 \,{\left ({\left ({\left (\frac{{\left (a^{6} b^{3} f - 4 \, a^{5} b^{4} f + 6 \, a^{4} b^{5} f - 4 \, a^{3} b^{6} f + a^{2} b^{7} f\right )} e^{\left (2 \, f x + 2 \, e\right )}}{a^{8} b^{2} f^{2} - 4 \, a^{7} b^{3} f^{2} + 6 \, a^{6} b^{4} f^{2} - 4 \, a^{5} b^{5} f^{2} + a^{4} b^{6} f^{2}} + \frac{3 \,{\left (2 \, a^{7} b^{2} f - 9 \, a^{6} b^{3} f + 16 \, a^{5} b^{4} f - 14 \, a^{4} b^{5} f + 6 \, a^{3} b^{6} f - a^{2} b^{7} f\right )}}{a^{8} b^{2} f^{2} - 4 \, a^{7} b^{3} f^{2} + 6 \, a^{6} b^{4} f^{2} - 4 \, a^{5} b^{5} f^{2} + a^{4} b^{6} f^{2}}\right )} e^{\left (2 \, f x + 2 \, e\right )} - \frac{3 \,{\left (2 \, a^{7} b^{2} f - 9 \, a^{6} b^{3} f + 16 \, a^{5} b^{4} f - 14 \, a^{4} b^{5} f + 6 \, a^{3} b^{6} f - a^{2} b^{7} f\right )}}{a^{8} b^{2} f^{2} - 4 \, a^{7} b^{3} f^{2} + 6 \, a^{6} b^{4} f^{2} - 4 \, a^{5} b^{5} f^{2} + a^{4} b^{6} f^{2}}\right )} e^{\left (2 \, f x + 2 \, e\right )} - \frac{a^{6} b^{3} f - 4 \, a^{5} b^{4} f + 6 \, a^{4} b^{5} f - 4 \, a^{3} b^{6} f + a^{2} b^{7} f}{a^{8} b^{2} f^{2} - 4 \, a^{7} b^{3} f^{2} + 6 \, a^{6} b^{4} f^{2} - 4 \, a^{5} b^{5} f^{2} + a^{4} b^{6} f^{2}}\right )}}{3 \,{\left (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\right )}^{\frac{3}{2}}} + \frac{2}{3 \, a^{2} \sqrt{b} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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