Integrand size = 25, antiderivative size = 154 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=-\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 a^4 d \sqrt {e \cos (c+d x)}}-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4+a^4 \sin (c+d x)\right )} \]
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Time = 0.15 (sec) , antiderivative size = 154, 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 = {2759, 2760, 2762, 2721, 2720} \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=-\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 a^4 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a \sin (c+d x)+a)^3} \]
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Rule 2720
Rule 2721
Rule 2759
Rule 2760
Rule 2762
Rubi steps \begin{align*} \text {integral}& = -\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}-\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx}{11 a^2} \\ & = -\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {\left (3 e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))} \, dx}{77 a^3} \\ & = -\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{77 a^4} \\ & = -\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\left (e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 a^4 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 a^4 d \sqrt {e \cos (c+d x)}}-\frac {4 e \sqrt {e \cos (c+d x)}}{11 a d (a+a \sin (c+d x))^3}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{77 d \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 = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=-\frac {(e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {15}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{10\ 2^{3/4} a^4 d e (1+\sin (c+d x))^{5/4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(582\) vs. \(2(162)=324\).
Time = 9.14 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.79
method | result | size |
default | \(\frac {2 \left (32 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-80 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+176 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-40 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-176 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-78 \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 )+176 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{2}}{77 \left (32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{4} \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}\) | \(583\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.49 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=\frac {{\left (3 i \, \sqrt {2} e \cos \left (d x + c\right )^{2} + {\left (i \, \sqrt {2} e \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} e\right )} \sin \left (d x + c\right ) - 4 i \, \sqrt {2} e\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (-3 i \, \sqrt {2} e \cos \left (d x + c\right )^{2} + {\left (-i \, \sqrt {2} e \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} e\right )} \sin \left (d x + c\right ) + 4 i \, \sqrt {2} e\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (e \cos \left (d x + c\right )^{2} - 3 \, e \sin \left (d x + c\right ) + 11 \, e\right )} \sqrt {e \cos \left (d x + c\right )}}{77 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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