Integrand size = 25, antiderivative size = 191 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=-\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )} \]
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Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2760, 2762, 2721, 2719} \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4 \sin (c+d x)+a^4\right )}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4} \]
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Rule 2719
Rule 2721
Rule 2760
Rule 2762
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}+\frac {5 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx}{13 a} \\ & = -\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}+\frac {5 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx}{39 a^2} \\ & = -\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {\int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{39 a^3} \\ & = -\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\int \sqrt {e \cos (c+d x)} \, dx}{39 a^4} \\ & = -\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{39 a^4 \sqrt {\cos (c+d x)}} \\ & = -\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^4 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{13 d e (a+a \sin (c+d x))^4}-\frac {10 (e \cos (c+d x))^{3/2}}{117 a d e (a+a \sin (c+d x))^3}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {2 (e \cos (c+d x))^{3/2}}{39 d e \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 = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=-\frac {(e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {17}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{12 \sqrt [4]{2} a^4 d e (1+\sin (c+d x))^{3/4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(693\) vs. \(2(195)=390\).
Time = 11.76 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.63
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.47 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=\frac {3 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 4 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} \cos \left (d x + c\right ) - 8 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) - 4 i \, \sqrt {2} \cos \left (d x + c\right ) - 8 i \, \sqrt {2}\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 ) + 3 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 4 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} \cos \left (d x + c\right ) + 8 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) + 4 i \, \sqrt {2} \cos \left (d x + c\right ) + 8 i \, \sqrt {2}\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 (3 \, \cos \left (d x + c\right )^{4} + 12 \, \cos \left (d x + c\right )^{3} - 14 \, \cos \left (d x + c\right )^{2} + {\left (3 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )^{2} - 23 \, \cos \left (d x + c\right ) + 9\right )} \sin \left (d x + c\right ) - 32 \, \cos \left (d x + c\right ) - 9\right )} \sqrt {e \cos \left (d x + c\right )}}{117 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 3 \, a^{4} d \cos \left (d x + c\right )^{3} - 8 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + 8 \, a^{4} d - {\left (a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d \cos \left (d x + c\right ) - 8 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx=\int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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