Optimal. Leaf size=92 \[ -\frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a-a \cosh (c+d x)}}-\frac{16 a^2 \sinh (c+d x) \sqrt{a-a \cosh (c+d x)}}{15 d}-\frac{2 a \sinh (c+d x) (a-a \cosh (c+d x))^{3/2}}{5 d} \]
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Rubi [A] time = 0.0520059, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2647, 2646} \[ -\frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a-a \cosh (c+d x)}}-\frac{16 a^2 \sinh (c+d x) \sqrt{a-a \cosh (c+d x)}}{15 d}-\frac{2 a \sinh (c+d x) (a-a \cosh (c+d x))^{3/2}}{5 d} \]
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 = 0.132919, size = 72, normalized size = 0.78 \[ \frac{a^2 \left (150 \cosh \left (\frac{1}{2} (c+d x)\right )-25 \cosh \left (\frac{3}{2} (c+d x)\right )+3 \cosh \left (\frac{5}{2} (c+d x)\right )\right ) \text{csch}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cosh (c+d x)}}{30 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 71, normalized size = 0.8 \begin{align*} -{\frac{16\,{a}^{3}}{15\,d}\sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 3\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+8 \right ){\frac{1}{\sqrt{-2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61936, size = 257, normalized size = 2.79 \begin{align*} \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-d x - c\right )}}{12 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, d x - 3 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} + \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, d x - 4 \, c\right )}}{12 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{\sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, d x - 5 \, c\right )}}{20 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{\sqrt{2} a^{\frac{5}{2}}}{20 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8034, size = 857, normalized size = 9.32 \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 [B] time = 1.18001, size = 262, normalized size = 2.85 \begin{align*} -\frac{\sqrt{2}{\left (3 \, \sqrt{-a e^{\left (d x + c\right )}} a^{2} e^{\left (2 \, d x + 2 \, c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 25 \, \sqrt{-a e^{\left (d x + c\right )}} a^{2} e^{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 150 \, \sqrt{-a e^{\left (d x + c\right )}} a^{2} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - \frac{{\left (150 \, a^{5} e^{\left (2 \, d x + 2 \, c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 25 \, a^{5} e^{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, a^{5} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{\sqrt{-a e^{\left (d x + c\right )}} a^{2}}\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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