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 = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 2646
Rule 2647
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*}
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Mathematica [A] time = 0.13, size = 71, normalized size = 0.80 \[ \frac {a^2 \left (150 \sinh \left (\frac {1}{2} (c+d x)\right )+25 \sinh \left (\frac {3}{2} (c+d x)\right )+3 \sinh \left (\frac {5}{2} (c+d x)\right )\right ) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cosh (c+d x)+1)}}{30 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 327, normalized size = 3.67 \[ \frac {\sqrt {\frac {1}{2}} {\left (3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \sinh \left (d x + c\right )^{5} + 25 \, a^{2} \cosh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{3} + 5 \, {\left (3 \, a^{2} \cosh \left (d x + c\right ) + 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} - 150 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \, a^{2} \cosh \left (d x + c\right ) + 15 \, a^{2}\right )} \sinh \left (d x + c\right )^{3} - 25 \, a^{2} \cosh \left (d x + c\right ) + 30 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 5 \, a^{2} \cosh \left (d x + c\right )^{2} + 15 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, a^{2} + 5 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{4} + 20 \, a^{2} \cosh \left (d x + c\right )^{3} + 90 \, a^{2} \cosh \left (d x + c\right )^{2} - 60 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{30 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 105, normalized size = 1.18 \[ -\frac {\sqrt {2} {\left ({\left (150 \, a^{\frac {5}{2}} e^{\left (2 \, d x + \frac {5}{2} \, c\right )} + 25 \, a^{\frac {5}{2}} e^{\left (d x + \frac {3}{2} \, c\right )} + 3 \, a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, c\right )}\right )} e^{\left (-\frac {5}{2} \, d x - 3 \, c\right )} - {\left (3 \, a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, d x + \frac {35}{2} \, c\right )} + 25 \, a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, d x + \frac {33}{2} \, c\right )} + 150 \, a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, d x + \frac {31}{2} \, c\right )}\right )} e^{\left (-15 \, c\right )}\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 73, normalized size = 0.82 \[ \frac {8 a^{3} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (3 \left (\cosh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right ) \sqrt {2}}{15 \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 121, normalized size = 1.36 \[ \frac {\sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}}{20 \, d} + \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}}{12 \, d} + \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} - \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{2 \, d} - \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {3}{2} \, d x - \frac {3}{2} \, c\right )}}{12 \, d} - \frac {\sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {5}{2} \, d x - \frac {5}{2} \, c\right )}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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