Integrand size = 25, antiderivative size = 127 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {6 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac {6 a^6 (e \cos (c+d x))^{3/2}}{5 d e^5 \left (a^3-a^3 \sin (c+d x)\right )} \]
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Time = 0.26 (sec) , antiderivative size = 127, 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 = {2749, 2759, 2762, 2721, 2719} \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}+\frac {6 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \sqrt {\cos (c+d x)}}-\frac {6 a^6 (e \cos (c+d x))^{3/2}}{5 d e^5 \left (a^3-a^3 \sin (c+d x)\right )} \]
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Rule 2719
Rule 2721
Rule 2749
Rule 2759
Rule 2762
Rubi steps \begin{align*} \text {integral}& = \frac {a^6 \int \frac {(e \cos (c+d x))^{5/2}}{(a-a \sin (c+d x))^3} \, dx}{e^6} \\ & = \frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac {\left (3 a^4\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a-a \sin (c+d x)} \, dx}{5 e^4} \\ & = \frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac {6 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))}+\frac {\left (3 a^3\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 e^4} \\ & = \frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac {6 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))}+\frac {\left (3 a^3 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 e^4 \sqrt {\cos (c+d x)}} \\ & = \frac {6 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac {6 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\frac {4\ 2^{3/4} a^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {3}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{5 d e (e \cos (c+d x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(139)=278\).
Time = 7.44 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.60
method | result | size |
default | \(-\frac {2 \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-20 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d}\) | \(330\) |
parts | \(\text {Expression too large to display}\) | \(797\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.50 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {3 \, {\left (-i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} + i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) + 2 i \, \sqrt {2} a^{3} + {\left (-i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) - 2 i \, \sqrt {2} a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} - i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) - 2 i \, \sqrt {2} a^{3} + {\left (i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) + 2 i \, \sqrt {2} a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} - {\left (3 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{5 \, {\left (d e^{4} \cos \left (d x + c\right )^{2} - d e^{4} \cos \left (d x + c\right ) - 2 \, d e^{4} + {\left (d e^{4} \cos \left (d x + c\right ) + 2 \, d e^{4}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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