Integrand size = 25, antiderivative size = 149 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 b^2\right ) e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e} \]
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Time = 0.11 (sec) , antiderivative size = 149, 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 = {2771, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\frac {2 e^2 \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 e \left (7 a^2+2 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e} \]
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {2}{7} \int (e \cos (c+d x))^{3/2} \left (\frac {7 a^2}{2}+b^2+\frac {9}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {1}{7} \left (7 a^2+2 b^2\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {1}{21} \left (\left (7 a^2+2 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {\left (\left (7 a^2+2 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 {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 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 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.77 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\frac {(e \cos (c+d x))^{3/2} \left (20 \left (7 a^2+2 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (-84 a b \cos (2 (c+d x))+5 \left (28 a^2+5 b^2\right ) \sin (c+d x)-3 b (28 a+5 b \sin (3 (c+d x)))\right )\right )}{210 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(157)=314\).
Time = 6.97 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.30
method | result | size |
default | \(\frac {2 e^{2} \left (240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-360 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+336 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -140 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+140 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-504 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +70 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-35 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+252 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-42 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a b \right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(343\) |
parts | \(-\frac {2 a^{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 (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 {4 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 )}{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 {4 a b \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}\) | \(424\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.96 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\frac {-5 i \, \sqrt {2} {\left (7 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} {\left (7 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (42 \, a b e \cos \left (d x + c\right )^{2} + 5 \, {\left (3 \, b^{2} e \cos \left (d x + c\right )^{2} - {\left (7 \, a^{2} + 2 \, b^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{105 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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