Optimal. Leaf size=71 \[ -\frac {8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt {a-a \cosh (x)}}-\frac {2}{15} a (5 A-3 B) \sinh (x) \sqrt {a-a \cosh (x)}+\frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2751, 2647, 2646} \[ -\frac {8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt {a-a \cosh (x)}}-\frac {2}{15} a (5 A-3 B) \sinh (x) \sqrt {a-a \cosh (x)}+\frac {2}{5} B \sinh (x) (a-a \cosh (x))^{3/2} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rubi steps
\begin {align*} \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx &=\frac {2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)-\frac {1}{5} (-5 A+3 B) \int (a-a \cosh (x))^{3/2} \, dx\\ &=-\frac {2}{15} a (5 A-3 B) \sqrt {a-a \cosh (x)} \sinh (x)+\frac {2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)+\frac {1}{15} (4 a (5 A-3 B)) \int \sqrt {a-a \cosh (x)} \, dx\\ &=-\frac {8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt {a-a \cosh (x)}}-\frac {2}{15} a (5 A-3 B) \sqrt {a-a \cosh (x)} \sinh (x)+\frac {2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 47, normalized size = 0.66 \[ -\frac {1}{15} a \coth \left (\frac {x}{2}\right ) \sqrt {a-a \cosh (x)} (2 (5 A-9 B) \cosh (x)-50 A+3 B \cosh (2 x)+39 B) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 279, normalized size = 3.93 \[ -\frac {\sqrt {\frac {1}{2}} {\left (3 \, B a \cosh \relax (x)^{5} + 3 \, B a \sinh \relax (x)^{5} + 5 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x)^{4} - 30 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x)^{3} + 5 \, {\left (3 \, B a \cosh \relax (x) + {\left (2 \, A - 3 \, B\right )} a\right )} \sinh \relax (x)^{4} - 30 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x)^{2} + 10 \, {\left (3 \, B a \cosh \relax (x)^{2} + 2 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x) - 3 \, {\left (3 \, A - 2 \, B\right )} a\right )} \sinh \relax (x)^{3} + 5 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x) + 30 \, {\left (B a \cosh \relax (x)^{3} + {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x)^{2} - 3 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x) - {\left (3 \, A - 2 \, B\right )} a\right )} \sinh \relax (x)^{2} + 3 \, B a + 5 \, {\left (3 \, B a \cosh \relax (x)^{4} + 4 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x)^{3} - 18 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x)^{2} - 12 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x) + {\left (2 \, A - 3 \, B\right )} a\right )} \sinh \relax (x)\right )} \sqrt {-\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{30 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 212, normalized size = 2.99 \[ \frac {1}{60} \, \sqrt {2} {\left (\frac {{\left (90 \, A a^{4} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 60 \, B a^{4} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 10 \, A a^{4} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{4} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{4} \mathrm {sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-2 \, x\right )}}{\sqrt {-a e^{x}} a^{2}} + \frac {3 \, \sqrt {-a e^{x}} B a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 10 \, \sqrt {-a e^{x}} A a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 15 \, \sqrt {-a e^{x}} B a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 90 \, \sqrt {-a e^{x}} A a^{6} \mathrm {sgn}\left (-e^{x} + 1\right ) + 60 \, \sqrt {-a e^{x}} B a^{6} \mathrm {sgn}\left (-e^{x} + 1\right )}{a^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 55, normalized size = 0.77 \[ \frac {8 \sinh \left (\frac {x}{2}\right ) a^{2} \cosh \left (\frac {x}{2}\right ) \left (6 B \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (5 A -3 B \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-10 A +6 B \right )}{15 \sqrt {-2 a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 199, normalized size = 2.80 \[ \frac {1}{6} \, {\left (\frac {9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} + \frac {9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}}\right )} A + \frac {1}{20} \, B {\left (\frac {{\left (5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} - 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} + 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} + \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}}\right )} \]
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 \[ \int \left (A+B\,\mathrm {cosh}\relax (x)\right )\,{\left (a-a\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- a \left (\cosh {\relax (x )} - 1\right )\right )^{\frac {3}{2}} \left (A + B \cosh {\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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