Integrand size = 10, antiderivative size = 179 \[ \int (a+b \sinh (x))^{5/2} \, dx=\frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2 i \left (23 a^2-9 b^2\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{15 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {16 i a \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{15 \sqrt {a+b \sinh (x)}} \]
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Time = 0.17 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2735, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \sinh (x))^{5/2} \, dx=-\frac {16 i a \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{15 \sqrt {a+b \sinh (x)}}+\frac {2 i \left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{15 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)} \]
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Rule 2732
Rule 2734
Rule 2735
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{5} \int \sqrt {a+b \sinh (x)} \left (\frac {1}{2} \left (5 a^2-3 b^2\right )+4 a b \sinh (x)\right ) \, dx \\ & = \frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {4}{15} \int \frac {\frac {1}{4} a \left (15 a^2-17 b^2\right )+\frac {1}{4} b \left (23 a^2-9 b^2\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx \\ & = \frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{15} \left (23 a^2-9 b^2\right ) \int \sqrt {a+b \sinh (x)} \, dx-\frac {1}{15} \left (8 a \left (a^2+b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx \\ & = \frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {\left (\left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)}\right ) \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{15 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (8 a \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{15 \sqrt {a+b \sinh (x)}} \\ & = \frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2 i \left (23 a^2-9 b^2\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{15 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {16 i a \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{15 \sqrt {a+b \sinh (x)}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99 \[ \int (a+b \sinh (x))^{5/2} \, dx=\frac {2 \left (23 i a^3+23 a^2 b-9 i a b^2-9 b^3\right ) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}-16 i a \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}+b \cosh (x) \left (22 a^2-3 b^2+3 b^2 \cosh (2 x)+28 a b \sinh (x)\right )}{15 \sqrt {a+b \sinh (x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (203 ) = 406\).
Time = 2.23 (sec) , antiderivative size = 917, normalized size of antiderivative = 5.12
method | result | size |
default | \(\frac {\frac {16 i \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{3} b}{15}+\frac {16 i \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \,b^{3}}{15}+2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{4}+\frac {4 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b^{2}}{5}-\frac {6 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{4}}{5}-\frac {46 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{4}}{15}-\frac {28 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b^{2}}{15}+\frac {6 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{4}}{5}+\frac {2 b^{4} \sinh \left (x \right )^{4}}{5}+\frac {28 a \,b^{3} \sinh \left (x \right )^{3}}{15}+\frac {22 a^{2} b^{2} \sinh \left (x \right )^{2}}{15}+\frac {2 b^{4} \sinh \left (x \right )^{2}}{5}+\frac {28 a \,b^{3} \sinh \left (x \right )}{15}+\frac {22 a^{2} b^{2}}{15}}{b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(917\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.59 \[ \int (a+b \sinh (x))^{5/2} \, dx=-\frac {4 \, {\left (\sqrt {2} {\left (a^{3} + 33 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{3} + 33 \, a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{3} + 33 \, a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 12 \, {\left (\sqrt {2} {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (3 \, b^{3} \cosh \left (x\right )^{4} + 3 \, b^{3} \sinh \left (x\right )^{4} + 22 \, a b^{2} \cosh \left (x\right )^{3} + 22 \, a b^{2} \cosh \left (x\right ) + 2 \, {\left (6 \, b^{3} \cosh \left (x\right ) + 11 \, a b^{2}\right )} \sinh \left (x\right )^{3} - 3 \, b^{3} - 4 \, {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (x\right )^{2} + 33 \, a b^{2} \cosh \left (x\right ) - 46 \, a^{2} b + 18 \, b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (6 \, b^{3} \cosh \left (x\right )^{3} + 33 \, a b^{2} \cosh \left (x\right )^{2} + 11 \, a b^{2} - 4 \, {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{90 \, {\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2}\right )}} \]
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\[ \int (a+b \sinh (x))^{5/2} \, dx=\int \left (a + b \sinh {\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int (a+b \sinh (x))^{5/2} \, dx=\int { {\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (a+b \sinh (x))^{5/2} \, dx=\int { {\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a+b \sinh (x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \]
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