Integrand size = 23, antiderivative size = 100 \[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\frac {6 a e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {2 b (e \sin (c+d x))^{7/2}}{7 d e} \]
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Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2715, 2721, 2719} \[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\frac {6 a e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {2 b (e \sin (c+d x))^{7/2}}{7 d e} \]
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Rule 2715
Rule 2719
Rule 2721
Rule 2748
Rubi steps \begin{align*} \text {integral}& = \frac {2 b (e \sin (c+d x))^{7/2}}{7 d e}+a \int (e \sin (c+d x))^{5/2} \, dx \\ & = -\frac {2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {2 b (e \sin (c+d x))^{7/2}}{7 d e}+\frac {1}{5} \left (3 a e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx \\ & = -\frac {2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {2 b (e \sin (c+d x))^{7/2}}{7 d e}+\frac {\left (3 a e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 \sqrt {\sin (c+d x)}} \\ & = \frac {6 a e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {2 b (e \sin (c+d x))^{7/2}}{7 d e} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\frac {2 (e \sin (c+d x))^{5/2} \left (-21 a E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+\sin ^{\frac {3}{2}}(c+d x) \left (-7 a \cos (c+d x)+5 b \sin ^2(c+d x)\right )\right )}{35 d \sin ^{\frac {5}{2}}(c+d x)} \]
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Time = 3.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.71
method | result | size |
default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 e}-\frac {e^{3} a \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{4}\left (d x +c \right )\right )+2 \left (\sin ^{2}\left (d x +c \right )\right )\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(171\) |
parts | \(-\frac {a \,e^{3} \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{4}\left (d x +c \right )\right )+2 \left (\sin ^{2}\left (d x +c \right )\right )\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}\) | \(173\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.27 \[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\frac {21 i \, \sqrt {2} a \sqrt {-i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} a \sqrt {i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (5 \, b e^{2} \cos \left (d x + c\right )^{2} + 7 \, a e^{2} \cos \left (d x + c\right ) - 5 \, b e^{2}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )}{35 \, d} \]
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\[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\int \left (e \sin {\left (c + d x \right )}\right )^{\frac {5}{2}} \left (a + b \cos {\left (c + d x \right )}\right )\, dx \]
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\[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right ) \,d x \]
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