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 = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 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 = 0.14, 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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fricas [B] time = 0.51, size = 328, normalized size = 3.57 \[ \frac {\sqrt {\frac {1}{2}} {\left (3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \sinh \left (d x + c\right )^{5} - 25 \, a^{2} \cosh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{3} + 5 \, {\left (3 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - 10 \, a^{2} \cosh \left (d x + c\right ) + 15 \, a^{2}\right )} \sinh \left (d x + c\right )^{3} - 25 \, a^{2} \cosh \left (d x + c\right ) + 30 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} - 5 \, a^{2} \cosh \left (d x + c\right )^{2} + 15 \, a^{2} \cosh \left (d x + c\right ) + 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} + 5 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{4} - 20 \, a^{2} \cosh \left (d x + c\right )^{3} + 90 \, a^{2} \cosh \left (d x + c\right )^{2} + 60 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{30 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 189, normalized size = 2.05 \[ -\frac {\sqrt {2} {\left (3 \, \sqrt {-a} a^{2} e^{\left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 25 \, \sqrt {-a} a^{2} e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 150 \, \sqrt {-a} a^{2} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 150 \, \sqrt {-a} a^{2} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 25 \, \sqrt {-a} a^{2} e^{\left (-\frac {3}{2} \, d x - \frac {3}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, \sqrt {-a} a^{2} e^{\left (-\frac {5}{2} \, d x - \frac {5}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 71, normalized size = 0.77 \[ -\frac {16 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (3 \left (\sinh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right )}{15 \sqrt {-2 a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 190, normalized size = 2.07 \[ \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-d x - c\right )}}{12 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {5}{2}}} - \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {5}{2}}} - \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, d x - 3 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {5}{2}}} + \frac {5 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, d x - 4 \, c\right )}}{12 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {5}{2}}} - \frac {\sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, d x - 5 \, c\right )}}{20 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {5}{2}}} - \frac {\sqrt {2} a^{\frac {5}{2}}}{20 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {5}{2}}} \]
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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