Integrand size = 18, antiderivative size = 98 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {64 a^3 (7 A-5 B) \sinh (x)}{105 \sqrt {a-a \cosh (x)}}-\frac {16}{105} a^2 (7 A-5 B) \sqrt {a-a \cosh (x)} \sinh (x)-\frac {2}{35} a (7 A-5 B) (a-a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a-a \cosh (x))^{5/2} \sinh (x) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2830, 2726, 2725} \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {64 a^3 (7 A-5 B) \sinh (x)}{105 \sqrt {a-a \cosh (x)}}-\frac {16}{105} a^2 (7 A-5 B) \sinh (x) \sqrt {a-a \cosh (x)}-\frac {2}{35} a (7 A-5 B) \sinh (x) (a-a \cosh (x))^{3/2}+\frac {2}{7} B \sinh (x) (a-a \cosh (x))^{5/2} \]
[In]
[Out]
Rule 2725
Rule 2726
Rule 2830
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.62 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{210} a^2 \sqrt {a-a \cosh (x)} (1246 A-1040 B+(-392 A+505 B) \cosh (x)+6 (7 A-20 B) \cosh (2 x)+15 B \cosh (3 x)) \coth \left (\frac {x}{2}\right ) \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {16 \sinh \left (\frac {x}{2}\right ) a^{3} \cosh \left (\frac {x}{2}\right ) \left (30 B \sinh \left (\frac {x}{2}\right )^{6}+\left (21 A -15 B \right ) \sinh \left (\frac {x}{2}\right )^{4}+\left (-28 A +20 B \right ) \sinh \left (\frac {x}{2}\right )^{2}+56 A -40 B \right )}{105 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(69\) |
parts | \(-\frac {16 A \sinh \left (\frac {x}{2}\right ) a^{3} \cosh \left (\frac {x}{2}\right ) \left (3 \sinh \left (\frac {x}{2}\right )^{4}-4 \sinh \left (\frac {x}{2}\right )^{2}+8\right )}{15 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}-\frac {16 B \sinh \left (\frac {x}{2}\right ) a^{3} \cosh \left (\frac {x}{2}\right ) \left (6 \sinh \left (\frac {x}{2}\right )^{6}-3 \sinh \left (\frac {x}{2}\right )^{4}+4 \sinh \left (\frac {x}{2}\right )^{2}-8\right )}{21 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(96\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (82) = 164\).
Time = 0.26 (sec) , antiderivative size = 564, normalized size of antiderivative = 5.76 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {\sqrt {\frac {1}{2}} {\left (15 \, B a^{2} \cosh \left (x\right )^{7} + 15 \, B a^{2} \sinh \left (x\right )^{7} + 21 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right )^{6} - 35 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right )^{5} + 525 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right )^{4} + 21 \, {\left (5 \, B a^{2} \cosh \left (x\right ) + {\left (2 \, A - 5 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{6} + 525 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right )^{3} + 7 \, {\left (45 \, B a^{2} \cosh \left (x\right )^{2} + 18 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right ) - 5 \, {\left (10 \, A - 11 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{5} - 35 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right )^{2} + 35 \, {\left (15 \, B a^{2} \cosh \left (x\right )^{3} + 9 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right )^{2} - 5 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right ) + 15 \, {\left (4 \, A - 3 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{4} + 21 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right ) + 35 \, {\left (15 \, B a^{2} \cosh \left (x\right )^{4} + 12 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right )^{3} - 10 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right )^{2} + 60 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right ) + 15 \, {\left (4 \, A - 3 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{3} + 15 \, B a^{2} + 35 \, {\left (9 \, B a^{2} \cosh \left (x\right )^{5} + 9 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right )^{4} - 10 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right )^{3} + 90 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right )^{2} + 45 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right ) - {\left (10 \, A - 11 \, B\right )} a^{2}\right )} \sinh \left (x\right )^{2} + 7 \, {\left (15 \, B a^{2} \cosh \left (x\right )^{6} + 18 \, {\left (2 \, A - 5 \, B\right )} a^{2} \cosh \left (x\right )^{5} - 25 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right )^{4} + 300 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right )^{3} + 225 \, {\left (4 \, A - 3 \, B\right )} a^{2} \cosh \left (x\right )^{2} - 10 \, {\left (10 \, A - 11 \, B\right )} a^{2} \cosh \left (x\right ) + 3 \, {\left (2 \, A - 5 \, B\right )} a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{420 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \]
[In]
[Out]
Timed out. \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.94 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{60} \, {\left (\frac {25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} + \frac {25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {3 \, \sqrt {2} a^{\frac {5}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}}\right )} A + \frac {1}{168} \, B {\left (\frac {{\left (21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} - 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}} - \frac {7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} - 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} + 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {5}{2}}}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.01 \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{840} \, \sqrt {2} {\left (\frac {{\left (2100 \, A a^{6} e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 1575 \, B a^{6} e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 350 \, A a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 385 \, B a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 42 \, A a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 105 \, B a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{6} \mathrm {sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-3 \, x\right )}}{\sqrt {-a e^{x}} a^{3}} - \frac {15 \, \sqrt {-a e^{x}} B a^{9} e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 42 \, \sqrt {-a e^{x}} A a^{9} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 105 \, \sqrt {-a e^{x}} B a^{9} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 350 \, \sqrt {-a e^{x}} A a^{9} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 385 \, \sqrt {-a e^{x}} B a^{9} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 2100 \, \sqrt {-a e^{x}} A a^{9} \mathrm {sgn}\left (-e^{x} + 1\right ) - 1575 \, \sqrt {-a e^{x}} B a^{9} \mathrm {sgn}\left (-e^{x} + 1\right )}{a^{7}}\right )} \]
[In]
[Out]
Timed out. \[ \int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a-a\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \]
[In]
[Out]