Integrand size = 25, antiderivative size = 197 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e} \]
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Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2771, 2941, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\frac {2 a e^2 \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a e \left (7 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e} \]
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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))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {2}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (\frac {9 a^2}{2}+2 b^2+\frac {13}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {4}{63} \int (e \cos (c+d x))^{3/2} \left (\frac {9}{4} a \left (7 a^2+6 b^2\right )+\frac {1}{4} b \left (89 a^2+28 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {1}{7} \left (a \left (7 a^2+6 b^2\right )\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {1}{21} \left (a \left (7 a^2+6 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}+\frac {\left (a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \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 {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e} \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.78 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\frac {(e \cos (c+d x))^{3/2} \left (80 \left (7 a^3+6 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {2}{3} \sqrt {\cos (c+d x)} \left (-756 a^2 b-147 b^3-28 \left (27 a^2 b+4 b^3\right ) \cos (2 (c+d x))+35 b^3 \cos (4 (c+d x))+840 a^3 \sin (c+d x)+450 a b^2 \sin (c+d x)-270 a b^2 \sin (3 (c+d x))\right )\right )}{840 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(449\) vs. \(2(201)=402\).
Time = 7.45 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.28
method | result | size |
default | \(\frac {2 e^{2} \left (-1120 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+2160 \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}+2800 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-3240 \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}+1512 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -2296 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-420 \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}+1260 \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}-2268 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +644 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+210 \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}-90 \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}-105 \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}-90 \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}+1134 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +28 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-189 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -28 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{315 \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}\) | \(450\) |
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^{2} \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\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 )+\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 )\right )}{3 \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 {9}{2}}}{9}-\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} e^{2}}{5}\right )}{d \,e^{3}}-\frac {6 a^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5 e d}+\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^{2} \left (24 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\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 )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 \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}\) | \(466\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.86 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\frac {-15 i \, \sqrt {2} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (35 \, b^{3} e \cos \left (d x + c\right )^{4} - 63 \, {\left (3 \, a^{2} b + b^{3}\right )} e \cos \left (d x + c\right )^{2} - 15 \, {\left (9 \, a b^{2} e \cos \left (d x + c\right )^{2} - {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{315 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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