Integrand size = 23, antiderivative size = 124 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
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Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\frac {10 a e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d} \]
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+a \int (e \cos (c+d x))^{7/2} \, dx \\ & = -\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} \left (5 a e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{21} \left (5 a e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (5 a e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=-\frac {16 \sqrt [4]{2} a (e \cos (c+d x))^{9/2} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (1+\sin (c+d x))^{9/4}} \]
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Time = 5.84 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86
method | result | size |
parts | \(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{4} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}\) | \(231\) |
default | \(-\frac {2 a \,e^{4} \left (-224 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+144 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+560 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-216 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-560 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+280 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-70 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(249\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\frac {-15 i \, \sqrt {2} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (7 \, a e^{3} \cos \left (d x + c\right )^{4} - 3 \, {\left (3 \, a e^{3} \cos \left (d x + c\right )^{2} + 5 \, a e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{63 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )} \,d x } \]
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\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \]
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