Integrand size = 25, antiderivative size = 168 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e} \]
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Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2757, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\frac {130 a^2 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {130 a^2 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}+\frac {26 a^2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d} \]
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2757
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{11} (13 a) \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx \\ & = -\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{11} \left (13 a^2\right ) \int (e \cos (c+d x))^{7/2} \, dx \\ & = -\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{77} \left (65 a^2 e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{231} \left (65 a^2 e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {\left (65 a^2 e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{231 \sqrt {e \cos (c+d x)}} \\ & = -\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=-\frac {32 \sqrt [4]{2} a^2 (e \cos (c+d x))^{9/2} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (1+\sin (c+d x))^{9/4}} \]
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Time = 18.39 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.76
method | result | size |
default | \(\frac {2 a^{2} e^{4} \left (4032 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10080 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4928 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8208 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12320 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2232 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12320 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-924 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-6160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+498 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1540 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-154 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{693 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(295\) |
parts | \(-\frac {2 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{4} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {4 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{4} \left (672 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2352 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3312 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2400 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+922 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-159 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {4 a^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}\) | \(473\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.85 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\frac {-195 i \, \sqrt {2} a^{2} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} a^{2} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (154 \, a^{2} e^{3} \cos \left (d x + c\right )^{4} + 3 \, {\left (21 \, a^{2} e^{3} \cos \left (d x + c\right )^{4} - 39 \, a^{2} e^{3} \cos \left (d x + c\right )^{2} - 65 \, a^{2} e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{693 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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