Integrand size = 25, antiderivative size = 137 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d \sqrt {e \cos (c+d x)}}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e} \]
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Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2757, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=\frac {6 a^2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}+\frac {6 a^2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{7 d} \]
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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))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {1}{7} (9 a) \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx \\ & = -\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {1}{7} \left (9 a^2\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {1}{7} \left (3 a^2 e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {\left (3 a^2 e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 \sqrt {e \cos (c+d x)}} \\ & = -\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d \sqrt {e \cos (c+d x)}}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 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 = 0.48 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=-\frac {16 \sqrt [4]{2} a^2 (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{5/4}} \]
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Time = 5.94 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.48
method | result | size |
default | \(\frac {2 a^{2} e^{2} \left (80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+112 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+84 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(203\) |
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^{2} \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\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 )\right )}{3 \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^{2} \left (24 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\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 )+\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} \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}\) | \(425\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=\frac {-15 i \, \sqrt {2} a^{2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} a^{2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (14 \, a^{2} e \cos \left (d x + c\right )^{2} + 5 \, {\left (a^{2} e \cos \left (d x + c\right )^{2} - 3 \, a^{2} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{35 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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