Integrand size = 25, antiderivative size = 101 \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {6 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)}}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d} \]
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Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2761, 2715, 2721, 2719} \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a d \sqrt {\cos (c+d x)}}+\frac {2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d} \]
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Rule 2715
Rule 2719
Rule 2721
Rule 2761
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {e^2 \int (e \cos (c+d x))^{5/2} \, dx}{a} \\ & = \frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {\left (3 e^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a} \\ & = \frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {\left (3 e^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a \sqrt {\cos (c+d x)}} \\ & = \frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {6 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)}}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65 \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=-\frac {4\ 2^{3/4} (e \cos (c+d x))^{11/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {11}{4},\frac {15}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{11 a d e (1+\sin (c+d x))^{11/4}} \]
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Time = 3.68 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.14
method | result | size |
default | \(\frac {2 e^{5} \left (80 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-160 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+120 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(216\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\frac {21 i \, \sqrt {2} e^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} e^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (5 \, e^{4} \cos \left (d x + c\right )^{3} + 7 \, e^{4} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{35 \, a d} \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{a \sin \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{a \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
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