Integrand size = 10, antiderivative size = 124 \[ \int (a+b \cosh (x))^{3/2} \, dx=-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}}+\frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x) \]
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Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2735, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}}+\frac {2}{3} b \sinh (x) \sqrt {a+b \cosh (x)}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]
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Rule 2732
Rule 2734
Rule 2735
Rule 2740
Rule 2742
Rule 2831
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{3} \int \frac {\frac {1}{2} \left (3 a^2+b^2\right )+2 a b \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx \\ & = \frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {1}{3} (4 a) \int \sqrt {a+b \cosh (x)} \, dx+\frac {1}{3} \left (-a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx \\ & = \frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {\left (4 a \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {\left (\left (-a^2+b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{3 \sqrt {a+b \cosh (x)}} \\ & = -\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \sqrt {a+b \cosh (x)}}+\frac {2}{3} b \sqrt {a+b \cosh (x)} \sinh (x) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {-8 i a (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b (a+b \cosh (x)) \sinh (x)}{3 \sqrt {a+b \cosh (x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(465\) vs. \(2(144)=288\).
Time = 1.43 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.76
method | result | size |
default | \(\frac {2 \left (4 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{4} b^{2}+2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} a b +2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b^{2}+3 a^{2} \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+4 a b \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+b^{2} \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-8 \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a b \right ) \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{3 \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\) | \(466\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.14 \[ \int (a+b \cosh (x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a^{2} + 3 \, b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a^{2} + 3 \, b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 24 \, {\left (\sqrt {2} a b \cosh \left (x\right ) + \sqrt {2} a b \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} - 8 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) - 4 \, a b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{9 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )}} \]
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\[ \int (a+b \cosh (x))^{3/2} \, dx=\int \left (a + b \cosh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (a+b \cosh (x))^{3/2} \, dx=\int { {\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (a+b \cosh (x))^{3/2} \, dx=\int { {\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cosh (x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \]
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