Integrand size = 22, antiderivative size = 187 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \]
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Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3208, 3235, 3228, 3197, 2732, 3205, 2740} \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}+\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}} \]
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Rule 2732
Rule 2740
Rule 3197
Rule 3205
Rule 3208
Rule 3228
Rule 3235
Rubi steps \begin{align*} \text {integral}& = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {1}{45} \int \frac {-3+\frac {3}{2} \cos (d+e x)+\frac {5}{2} \sin (d+e x)}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx \\ & = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {1}{675} \int \frac {\frac {23}{2}+6 \cos (d+e x)+10 \sin (d+e x)}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx \\ & = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {2}{675} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx+\frac {1}{90} \int \frac {1}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx \\ & = -\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {2}{675} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )} \, dx+\frac {1}{90} \int \frac {1}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}} \, dx \\ & = \frac {4 \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 )}{675 e}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 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 = 2.47 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {-24 \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}+\frac {272}{3} \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}+\frac {100 (5+17 \sin (d+e x))}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}-\frac {10 (115+136 \sin (d+e x))}{3 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+23 \sqrt {\frac {10}{3}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}+\frac {20 \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {\frac {1}{17}+\frac {\cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {34}}}}-\frac {20 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{6750 e} \]
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Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.13
method | result | size |
default | \(\frac {17 \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}}\, \left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}+2 \sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \left (-\frac {\sqrt {34}\, \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}}}{765 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\frac {\sqrt {34}}{17}\right )^{2}}+\frac {136 \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}{675 \sqrt {-\cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2} \sqrt {34}\, \left (-289 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-17 \sqrt {34}\right )}}+\frac {46 \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{-\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 )}{675 \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}}}+\frac {8 \sqrt {34}\, \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{-\sqrt {34}+17}}\, \left (\left (-\frac {\sqrt {34}}{17}-1\right ) \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 )+\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 )\right )}{675 \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 )}{2 \left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\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}\) | \(586\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {53 \, \sqrt {5 i + 3} {\left (\left (48 i + 80\right ) \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (-\left (9 i + 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) - \left (6 i + 10\right ) \, \sqrt {2}\right )} \sin \left (e x + d\right ) - \left (36 i + 60\right ) \, \sqrt {2} \cos \left (e x + d\right ) - \left (87 i + 145\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 ) + 53 \, \sqrt {-5 i + 3} {\left (-\left (48 i - 80\right ) \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (\left (9 i - 15\right ) \, \sqrt {2} \cos \left (e x + d\right ) + \left (6 i - 10\right ) \, \sqrt {2}\right )} \sin \left (e x + d\right ) + \left (36 i - 60\right ) \, \sqrt {2} \cos \left (e x + d\right ) + \left (87 i - 145\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 ) + 408 \, \sqrt {5 i + 3} {\left (-16 i \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (3 i \, \sqrt {2} \cos \left (e x + d\right ) + 2 i \, \sqrt {2}\right )} \sin \left (e x + d\right ) + 12 i \, \sqrt {2} \cos \left (e x + d\right ) + 29 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 ) + 408 \, \sqrt {-5 i + 3} {\left (16 i \, \sqrt {2} \cos \left (e x + d\right )^{2} + 10 \, {\left (-3 i \, \sqrt {2} \cos \left (e x + d\right ) - 2 i \, \sqrt {2}\right )} \sin \left (e x + d\right ) - 12 i \, \sqrt {2} \cos \left (e x + d\right ) - 29 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 ) - 204 \, {\left (120 \, \cos \left (e x + d\right )^{2} + {\left (64 \, \cos \left (e x + d\right ) + 21\right )} \sin \left (e x + d\right ) - 35 \, \cos \left (e x + d\right ) - 60\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{137700 \, {\left (16 \, e \cos \left (e x + d\right )^{2} - 12 \, e \cos \left (e x + d\right ) - 10 \, {\left (3 \, e \cos \left (e x + d\right ) + 2 \, e\right )} \sin \left (e x + d\right ) - 29 \, e\right )}} \]
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\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\int \frac {1}{\left (5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\int { \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\int \frac {1}{{\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{5/2}} \,d x \]
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