Optimal. Leaf size=94 \[ \frac {64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt {a \cosh (x)+a}}+\frac {16}{105} a^2 (7 A+5 B) \sinh (x) \sqrt {a \cosh (x)+a}+\frac {2}{35} a (7 A+5 B) \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2} \]
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Rubi [A] time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2751, 2647, 2646} \[ \frac {64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt {a \cosh (x)+a}}+\frac {16}{105} a^2 (7 A+5 B) \sinh (x) \sqrt {a \cosh (x)+a}+\frac {2}{35} a (7 A+5 B) \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rubi steps
\begin {align*} \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx &=\frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac {1}{7} (7 A+5 B) \int (a+a \cosh (x))^{5/2} \, dx\\ &=\frac {2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac {1}{35} (8 a (7 A+5 B)) \int (a+a \cosh (x))^{3/2} \, dx\\ &=\frac {16}{105} a^2 (7 A+5 B) \sqrt {a+a \cosh (x)} \sinh (x)+\frac {2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac {1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt {a+a \cosh (x)} \, dx\\ &=\frac {64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt {a+a \cosh (x)}}+\frac {16}{105} a^2 (7 A+5 B) \sqrt {a+a \cosh (x)} \sinh (x)+\frac {2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.13, size = 60, normalized size = 0.64 \[ \frac {1}{210} a^2 \tanh \left (\frac {x}{2}\right ) \sqrt {a (\cosh (x)+1)} ((392 A+505 B) \cosh (x)+6 (7 A+20 B) \cosh (2 x)+1246 A+15 B \cosh (3 x)+1040 B) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 563, normalized size = 5.99 \[ \frac {\sqrt {\frac {1}{2}} {\left (15 \, B a^{2} \cosh \relax (x)^{7} + 15 \, B a^{2} \sinh \relax (x)^{7} + 21 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x)^{6} + 35 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x)^{5} + 525 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x)^{4} + 21 \, {\left (5 \, B a^{2} \cosh \relax (x) + {\left (2 \, A + 5 \, B\right )} a^{2}\right )} \sinh \relax (x)^{6} - 525 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x)^{3} + 7 \, {\left (45 \, B a^{2} \cosh \relax (x)^{2} + 18 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x) + 5 \, {\left (10 \, A + 11 \, B\right )} a^{2}\right )} \sinh \relax (x)^{5} - 35 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x)^{2} + 35 \, {\left (15 \, B a^{2} \cosh \relax (x)^{3} + 9 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x)^{2} + 5 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x) + 15 \, {\left (4 \, A + 3 \, B\right )} a^{2}\right )} \sinh \relax (x)^{4} - 21 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x) + 35 \, {\left (15 \, B a^{2} \cosh \relax (x)^{4} + 12 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x)^{3} + 10 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x)^{2} + 60 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x) - 15 \, {\left (4 \, A + 3 \, B\right )} a^{2}\right )} \sinh \relax (x)^{3} - 15 \, B a^{2} + 35 \, {\left (9 \, B a^{2} \cosh \relax (x)^{5} + 9 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x)^{4} + 10 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x)^{3} + 90 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x)^{2} - 45 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x) - {\left (10 \, A + 11 \, B\right )} a^{2}\right )} \sinh \relax (x)^{2} + 7 \, {\left (15 \, B a^{2} \cosh \relax (x)^{6} + 18 \, {\left (2 \, A + 5 \, B\right )} a^{2} \cosh \relax (x)^{5} + 25 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x)^{4} + 300 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x)^{3} - 225 \, {\left (4 \, A + 3 \, B\right )} a^{2} \cosh \relax (x)^{2} - 10 \, {\left (10 \, A + 11 \, B\right )} a^{2} \cosh \relax (x) - 3 \, {\left (2 \, A + 5 \, B\right )} a^{2}\right )} \sinh \relax (x)\right )} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{420 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 153, normalized size = 1.63 \[ -\frac {1}{840} \, \sqrt {2} {\left (\frac {{\left (2100 \, A a^{6} e^{\left (3 \, x\right )} + 1575 \, B a^{6} e^{\left (3 \, x\right )} + 350 \, A a^{6} e^{\left (2 \, x\right )} + 385 \, B a^{6} e^{\left (2 \, x\right )} + 42 \, A a^{6} e^{x} + 105 \, B a^{6} e^{x} + 15 \, B a^{6}\right )} e^{\left (-\frac {7}{2} \, x\right )}}{a^{\frac {7}{2}}} - \frac {15 \, B a^{\frac {19}{2}} e^{\left (\frac {7}{2} \, x\right )} + 42 \, A a^{\frac {19}{2}} e^{\left (\frac {5}{2} \, x\right )} + 105 \, B a^{\frac {19}{2}} e^{\left (\frac {5}{2} \, x\right )} + 350 \, A a^{\frac {19}{2}} e^{\left (\frac {3}{2} \, x\right )} + 385 \, B a^{\frac {19}{2}} e^{\left (\frac {3}{2} \, x\right )} + 2100 \, A a^{\frac {19}{2}} e^{\left (\frac {1}{2} \, x\right )} + 1575 \, B a^{\frac {19}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{7}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 71, normalized size = 0.76 \[ \frac {8 \cosh \left (\frac {x}{2}\right ) a^{3} \sinh \left (\frac {x}{2}\right ) \left (30 B \left (\sinh ^{6}\left (\frac {x}{2}\right )\right )+\left (21 A +105 B \right ) \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (70 A +140 B \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+105 A +105 B \right ) \sqrt {2}}{105 \sqrt {a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 237, normalized size = 2.52 \[ \frac {1}{60} \, {\left (3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, x\right )} + 25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, x\right )} + 150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, x\right )} - 150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {1}{2} \, x\right )} - 25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {3}{2} \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {5}{2} \, x\right )}\right )} A + \frac {1}{168} \, {\left ({\left (3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} + 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac {9}{2} \, x\right )} + {\left (7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} + 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac {5}{2} \, x\right )}\right )} B \]
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 \[ \int \left (A+B\,\mathrm {cosh}\relax (x)\right )\,{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{5/2} \,d x \]
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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