Integrand size = 22, antiderivative size = 94 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=-\frac {\sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \]
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Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3207, 3197, 2732} \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=-\frac {\sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}} \]
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Rule 2732
Rule 3197
Rule 3207
Rubi steps \begin{align*} \text {integral}& = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-\frac {1}{30} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx \\ & = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-\frac {1}{30} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \, dx \\ & = -\frac {\sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 4.79 (sec) , antiderivative size = 390, normalized size of antiderivative = 4.15 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\frac {18 \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}-68 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}+\frac {20 (5+17 \sin (d+e x))}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}-2 \sqrt {30} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}-\frac {15 \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {\frac {1}{17}+\frac {\cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {34}}}}+\frac {15 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{450 e} \]
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Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 425, normalized size of antiderivative = 4.52
method | result | size |
default | \(\frac {\sqrt {34}\, \left (255 \sqrt {\left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right ) \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}\, \sqrt {\frac {\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+1}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \operatorname {EllipticF}\left (\sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}, i \sqrt {\frac {\sqrt {34}+17}{-\sqrt {34}+17}}\right )-255 \sqrt {\left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right ) \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}\, \sqrt {\frac {\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+1}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \operatorname {EllipticE}\left (\sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}, i \sqrt {\frac {\sqrt {34}+17}{-\sqrt {34}+17}}\right )+289 \sqrt {\left (\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}-289 \sqrt {\left (\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\right ) \sqrt {17}}{4335 \sqrt {\left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right ) \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right ) \sqrt {\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2}\, e}\) | \(425\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\frac {\sqrt {5 i + 3} {\left (-\left (9 i + 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) - \left (15 i + 25\right ) \, \sqrt {2} \sin \left (e x + d\right ) - \left (6 i + 10\right ) \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right ) + \sqrt {-5 i + 3} {\left (\left (9 i - 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) + \left (15 i - 25\right ) \, \sqrt {2} \sin \left (e x + d\right ) + \left (6 i - 10\right ) \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right ) - 51 \, \sqrt {5 i + 3} {\left (-3 i \, \sqrt {2} \cos \left (e x + d\right ) - 5 i \, \sqrt {2} \sin \left (e x + d\right ) - 2 i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right )\right ) - 51 \, \sqrt {-5 i + 3} {\left (3 i \, \sqrt {2} \cos \left (e x + d\right ) + 5 i \, \sqrt {2} \sin \left (e x + d\right ) + 2 i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right )\right ) - 102 \, {\left (5 \, \cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right )\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{1530 \, {\left (3 \, e \cos \left (e x + d\right ) + 5 \, e \sin \left (e x + d\right ) + 2 \, e\right )}} \]
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\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int \frac {1}{\left (5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx=\int \frac {1}{{\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{3/2}} \,d x \]
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