Integrand size = 25, antiderivative size = 188 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2771, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\frac {10 e^4 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 e^3 \left (11 a^2+2 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}+\frac {2 e \left (11 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e} \]
[In]
[Out]
Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{7/2} \left (\frac {11 a^2}{2}+b^2+\frac {13}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {1}{11} \left (11 a^2+2 b^2\right ) \int (e \cos (c+d x))^{7/2} \, dx \\ & = -\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {1}{77} \left (5 \left (11 a^2+2 b^2\right ) e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {1}{231} \left (5 \left (11 a^2+2 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}+\frac {\left (5 \left (11 a^2+2 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 {26 a b (e \cos (c+d x))^{9/2}}{99 d e}+\frac {10 \left (11 a^2+2 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 \left (11 a^2+2 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (11 a^2+2 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e} \\ \end{align*}
Time = 1.73 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.85 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\frac {(e \cos (c+d x))^{7/2} \left (-154 a b \sqrt {\cos (c+d x)}+40 \left (11 a^2+2 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{6} \sqrt {\cos (c+d x)} \left (6 \left (572 a^2+41 b^2\right ) \sin (c+d x)-14 b \cos (4 (c+d x)) (22 a+9 b \sin (c+d x))+8 \cos (2 (c+d x)) \left (-154 a b+9 \left (11 a^2-5 b^2\right ) \sin (c+d x)\right )\right )\right )}{924 d \cos ^{\frac {7}{2}}(c+d x)} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(471\) vs. \(2(192)=384\).
Time = 16.84 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.51
method | result | size |
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^{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 {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^{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 )}{231 \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 {9}{2}}}{9 d e}\) | \(472\) |
default | \(\frac {2 e^{4} \left (4032 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-10080 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+4928 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -1584 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+9792 \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}-12320 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +2376 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-4608 \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}+12320 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -1848 \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}+924 \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}-6160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +528 \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}-30 \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}-165 \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}-30 \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}+1540 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-154 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a b \right )}{693 \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}\) | \(473\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.92 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\frac {-15 i \, \sqrt {2} {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (154 \, a b e^{3} \cos \left (d x + c\right )^{4} + 3 \, {\left (21 \, b^{2} e^{3} \cos \left (d x + c\right )^{4} - 3 \, {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2} - 5 \, {\left (11 \, a^{2} + 2 \, b^{2}\right )} e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{693 \, d} \]
[In]
[Out]
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
[In]
[Out]