Integrand size = 10, antiderivative size = 84 \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}} \]
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Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2743, 21, 2734, 2732} \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]
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Rule 21
Rule 2732
Rule 2734
Rule 2743
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 \int \frac {-\frac {a}{2}-\frac {1}{2} b \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{a^2-b^2} \\ & = -\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\int \sqrt {a+b \cosh (x)} \, dx}{a^2-b^2} \\ & = -\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}} \\ & = -\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 \left (i (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+b \sinh (x)\right )}{(a-b) (a+b) \sqrt {a+b \cosh (x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(96)=192\).
Time = 1.18 (sec) , antiderivative size = 298, normalized size of antiderivative = 3.55
method | result | size |
default | \(-\frac {2 \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) b +2 \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) b \right )}{\sqrt {-\frac {2 b}{a -b}}\, \left (a -b \right ) \left (a +b \right ) \sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\) | \(298\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.92 \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} a b \cosh \left (x\right )^{2} + \sqrt {2} a b \sinh \left (x\right )^{2} + 2 \, \sqrt {2} a^{2} \cosh \left (x\right ) + \sqrt {2} a b + 2 \, {\left (\sqrt {2} a b \cosh \left (x\right ) + \sqrt {2} a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 3 \, {\left (\sqrt {2} b^{2} \cosh \left (x\right )^{2} + \sqrt {2} b^{2} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} a b \cosh \left (x\right ) + \sqrt {2} b^{2} + 2 \, {\left (\sqrt {2} b^{2} \cosh \left (x\right ) + \sqrt {2} a b\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 6 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + a b \cosh \left (x\right ) + {\left (2 \, b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}\right )}}{3 \, {\left (a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{3} b - a b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b - a b^{3} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
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\[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \cosh {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \]
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