Integrand size = 25, antiderivative size = 237 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e} \]
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Time = 0.23 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2771, 2941, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\frac {10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rule 2941
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {2}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (\frac {13 a^2}{2}+2 b^2+\frac {17}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {4}{143} \int (e \cos (c+d x))^{7/2} \left (\frac {13}{4} a \left (11 a^2+6 b^2\right )+\frac {1}{4} b \left (177 a^2+44 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{11} \left (a \left (11 a^2+6 b^2\right )\right ) \int (e \cos (c+d x))^{7/2} \, dx \\ & = -\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{77} \left (5 a \left (11 a^2+6 b^2\right ) e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{231} \left (5 a \left (11 a^2+6 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {\left (5 a \left (11 a^2+6 b^2\right ) e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{231 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e} \\ \end{align*}
Time = 1.94 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.86 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\frac {(e \cos (c+d x))^{7/2} \left (-154 b \left (78 a^2+11 b^2\right ) \sqrt {\cos (c+d x)}+2080 \left (11 a^3+6 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{3} \sqrt {\cos (c+d x)} \left (-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+154 b \left (-78 a^2+b^2\right ) \cos (4 (c+d x))+693 b^3 \cos (6 (c+d x))+156 a \left (506 a^2+213 b^2\right ) \sin (c+d x)+234 a \left (44 a^2-39 b^2\right ) \sin (3 (c+d x))-4914 a b^2 \sin (5 (c+d x))\right )\right )}{48048 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(513\) vs. \(2(237)=474\).
Time = 40.98 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.17
method | result | size |
parts | \(-\frac {2 a^{3} \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 b^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\right )}{d \,e^{3}}+\frac {4 a \,b^{2} \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 (672 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2352 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3312 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2400 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+922 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-159 \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 )+5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 \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^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{3 d e}\) | \(514\) |
default | \(\frac {2 e^{4} \left (-1170 \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 ) a \,b^{2}-3003 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +30888 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-393120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+381888 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-179712 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+36036 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-1170 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-2145 \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 ) a^{3}-20592 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-433664 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+157248 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+310464 \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-120120 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +96096 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +30030 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -308 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}-88704 \left (\sin ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+6864 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-24024 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+240240 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +308000 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-113960 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+18172 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+308 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-240240 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b \right )}{9009 \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}\) | \(618\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.85 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\frac {-195 i \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (693 \, b^{3} e^{3} \cos \left (d x + c\right )^{6} - 1001 \, {\left (3 \, a^{2} b + b^{3}\right )} e^{3} \cos \left (d x + c\right )^{4} - 39 \, {\left (63 \, a b^{2} e^{3} \cos \left (d x + c\right )^{4} - 3 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2} - 5 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{9009 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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