Optimal. Leaf size=89 \[ \frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a \cosh (c+d x)+a}}+\frac{16 a^2 \sinh (c+d x) \sqrt{a \cosh (c+d x)+a}}{15 d}+\frac{2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.0468435, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a \cosh (c+d x)+a}}+\frac{16 a^2 \sinh (c+d x) \sqrt{a \cosh (c+d x)+a}}{15 d}+\frac{2 a \sinh (c+d x) (a \cosh (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cosh (c+d x))^{5/2} \, dx &=\frac{2 a (a+a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d}+\frac{1}{5} (8 a) \int (a+a \cosh (c+d x))^{3/2} \, dx\\ &=\frac{16 a^2 \sqrt{a+a \cosh (c+d x)} \sinh (c+d x)}{15 d}+\frac{2 a (a+a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d}+\frac{1}{15} \left (32 a^2\right ) \int \sqrt{a+a \cosh (c+d x)} \, dx\\ &=\frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a+a \cosh (c+d x)}}+\frac{16 a^2 \sqrt{a+a \cosh (c+d x)} \sinh (c+d x)}{15 d}+\frac{2 a (a+a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.12481, size = 71, normalized size = 0.8 \[ \frac{a^2 \left (150 \sinh \left (\frac{1}{2} (c+d x)\right )+25 \sinh \left (\frac{3}{2} (c+d x)\right )+3 \sinh \left (\frac{5}{2} (c+d x)\right )\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cosh (c+d x)+1)}}{30 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 73, normalized size = 0.8 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{15\,d}\cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 3\, \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+4\, \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+8 \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59718, size = 163, normalized size = 1.83 \begin{align*} \frac{\sqrt{2} a^{\frac{5}{2}} e^{\left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )}}{20 \, d} + \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )}}{12 \, d} + \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} - \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{2 \, d} - \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-\frac{3}{2} \, d x - \frac{3}{2} \, c\right )}}{12 \, d} - \frac{\sqrt{2} a^{\frac{5}{2}} e^{\left (-\frac{5}{2} \, d x - \frac{5}{2} \, c\right )}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8515, size = 856, normalized size = 9.62 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \sinh \left (d x + c\right )^{5} + 25 \, a^{2} \cosh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{3} + 5 \,{\left (3 \, a^{2} \cosh \left (d x + c\right ) + 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} - 150 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \,{\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \, a^{2} \cosh \left (d x + c\right ) + 15 \, a^{2}\right )} \sinh \left (d x + c\right )^{3} - 25 \, a^{2} \cosh \left (d x + c\right ) + 30 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} + 5 \, a^{2} \cosh \left (d x + c\right )^{2} + 15 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, a^{2} + 5 \,{\left (3 \, a^{2} \cosh \left (d x + c\right )^{4} + 20 \, a^{2} \cosh \left (d x + c\right )^{3} + 90 \, a^{2} \cosh \left (d x + c\right )^{2} - 60 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt{\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{30 \,{\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\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 [A] time = 1.19484, size = 142, normalized size = 1.6 \begin{align*} -\frac{\sqrt{2}{\left ({\left (150 \, a^{\frac{5}{2}} e^{\left (2 \, d x + \frac{5}{2} \, c\right )} + 25 \, a^{\frac{5}{2}} e^{\left (d x + \frac{3}{2} \, c\right )} + 3 \, a^{\frac{5}{2}} e^{\left (\frac{1}{2} \, c\right )}\right )} e^{\left (-\frac{5}{2} \, d x - 3 \, c\right )} -{\left (3 \, a^{\frac{5}{2}} e^{\left (\frac{5}{2} \, d x + \frac{35}{2} \, c\right )} + 25 \, a^{\frac{5}{2}} e^{\left (\frac{3}{2} \, d x + \frac{33}{2} \, c\right )} + 150 \, a^{\frac{5}{2}} e^{\left (\frac{1}{2} \, d x + \frac{31}{2} \, c\right )}\right )} e^{\left (-15 \, c\right )}\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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