Integrand size = 25, antiderivative size = 143 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {14 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \]
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Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2762, 2716, 2721, 2719} \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}} \]
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Rule 2716
Rule 2719
Rule 2721
Rule 2762
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{7/2}} \, dx}{9 a} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}+\frac {7 \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx}{15 a e^2} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}-\frac {7 \int \sqrt {e \cos (c+d x)} \, dx}{15 a e^4} \\ & = \frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))}-\frac {\left (7 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 a e^4 \sqrt {\cos (c+d x)}} \\ & = -\frac {14 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \\ \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 = 0.46 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {13}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{10 \sqrt [4]{2} a d e (e \cos (c+d x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(151)=302\).
Time = 6.22 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.41
method | result | size |
default | \(\frac {\frac {448 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}-\frac {224 E\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}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-\frac {896 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {448 E\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}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {2128 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{45}-\frac {112 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {784 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{45}+\frac {112 \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 ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {44 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}-\frac {14 \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 )}{15}-\frac {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{9}}{\left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) 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}\, e^{3} d}\) | \(488\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{2} - 7 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) - 2\right )} \sqrt {e \cos \left (d x + c\right )}}{45 \, {\left (a d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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