Integrand size = 25, antiderivative size = 225 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=-\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )} \]
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Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2760, 2762, 2716, 2721, 2719} \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}-\frac {14}{221 d e \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}-\frac {18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}} \]
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Rule 2716
Rule 2719
Rule 2721
Rule 2760
Rule 2762
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}+\frac {9 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx}{17 a} \\ & = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}+\frac {63 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx}{221 a^2} \\ & = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {35 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx}{221 a^3} \\ & = -\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}+\frac {21 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{221 a^4} \\ & = \frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac {21 \int \sqrt {e \cos (c+d x)} \, dx}{221 a^4 e^2} \\ & = \frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (21 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{221 a^4 e^2 \sqrt {\cos (c+d x)}} \\ & = -\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{221 a^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {42 \sin (c+d x)}{221 a^4 d e \sqrt {e \cos (c+d x)}}-\frac {2}{17 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac {18}{221 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {14}{221 d e \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.29 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {21}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{8 \sqrt [4]{2} a^4 d e \sqrt {e \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(877\) vs. \(2(225)=450\).
Time = 13.88 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.90
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.56 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\frac {21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{5} + 8 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 4 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 2 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 8 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{5} - 8 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 4 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 2 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 8 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (84 \, \cos \left (d x + c\right )^{4} - 224 \, \cos \left (d x + c\right )^{2} + {\left (21 \, \cos \left (d x + c\right )^{4} - 161 \, \cos \left (d x + c\right )^{2} + 117\right )} \sin \left (d x + c\right ) + 104\right )} \sqrt {e \cos \left (d x + c\right )}}{221 \, {\left (a^{4} d e^{2} \cos \left (d x + c\right )^{5} - 8 \, a^{4} d e^{2} \cos \left (d x + c\right )^{3} + 8 \, a^{4} d e^{2} \cos \left (d x + c\right ) - 4 \, {\left (a^{4} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{4} d e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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