Integrand size = 25, antiderivative size = 169 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2} \]
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Time = 0.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2759, 2761, 2715, 2721, 2720} \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {26 e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 a^3 d}+\frac {26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2} \]
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Rule 2715
Rule 2720
Rule 2721
Rule 2759
Rule 2761
Rubi steps \begin{align*} \text {integral}& = \frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^2\right ) \int \frac {(e \cos (c+d x))^{11/2}}{a+a \sin (c+d x)} \, dx}{5 a^2} \\ & = \frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^4\right ) \int (e \cos (c+d x))^{7/2} \, dx}{5 a^3} \\ & = \frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^6\right ) \int (e \cos (c+d x))^{3/2} \, dx}{7 a^3} \\ & = \frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^8\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 a^3} \\ & = \frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {\left (13 e^8 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^3 \sqrt {e \cos (c+d x)}} \\ & = \frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{17/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {17}{4},\frac {21}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{17 a^3 d e (1+\sin (c+d x))^{17/4}} \]
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Time = 185.84 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {2 e^{8} \left (1120 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2800 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3240 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+784 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+840 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1624 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1162 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+217 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(251\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {-195 i \, \sqrt {2} e^{\frac {15}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} e^{\frac {15}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (35 \, e^{7} \cos \left (d x + c\right )^{4} - 252 \, e^{7} \cos \left (d x + c\right )^{2} + 15 \, {\left (9 \, e^{7} \cos \left (d x + c\right )^{2} - 13 \, e^{7}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{315 \, a^{3} d} \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {15}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {15}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{15/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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