Integrand size = 34, antiderivative size = 98 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {1}{8} a^2 (4 A+B) c x-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3046, 2939, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {a^2 c (4 A+B) \cos ^3(e+f x)}{12 f}+\frac {a^2 c (4 A+B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^2 c x (4 A+B)-\frac {B c \cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2748
Rule 2939
Rule 3046
Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx \\ & = -\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} (a (4 A+B) c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx \\ & = -\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} \left (a^2 (4 A+B) c\right ) \int \cos ^2(e+f x) \, dx \\ & = -\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac {1}{8} \left (a^2 (4 A+B) c\right ) \int 1 \, dx \\ & = \frac {1}{8} a^2 (4 A+B) c x-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.17 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {a^2 c \cos (e+f x) \left (12 (4 A+B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} (8 A+8 B+8 (A+B) \cos (2 (e+f x))-3 (8 A+B) \sin (e+f x)+3 B \sin (3 (e+f x)))\right )}{48 f \sqrt {\cos ^2(e+f x)}} \]
[In]
[Out]
Time = 1.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(\frac {c \left (\frac {\left (-A -B \right ) \cos \left (3 f x +3 e \right )}{3}+A \sin \left (2 f x +2 e \right )-\frac {B \sin \left (4 f x +4 e \right )}{8}+\left (-A -B \right ) \cos \left (f x +e \right )+2 f x A +\frac {f x B}{2}-\frac {4 A}{3}-\frac {4 B}{3}\right ) a^{2}}{4 f}\) | \(82\) |
risch | \(\frac {a^{2} c x A}{2}+\frac {a^{2} c x B}{8}-\frac {a^{2} c \cos \left (f x +e \right ) A}{4 f}-\frac {a^{2} c \cos \left (f x +e \right ) B}{4 f}-\frac {B \,a^{2} c \sin \left (4 f x +4 e \right )}{32 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right ) A}{12 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right ) B}{12 f}+\frac {A \,a^{2} c \sin \left (2 f x +2 e \right )}{4 f}\) | \(126\) |
parts | \(-\frac {\left (-A \,a^{2} c -B \,a^{2} c \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (-A \,a^{2} c +B \,a^{2} c \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (A \,a^{2} c +B \,a^{2} c \right ) \cos \left (f x +e \right )}{f}+a^{2} c x A -\frac {B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(152\) |
derivativedivides | \(\frac {\frac {A \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \cos \left (f x +e \right ) a^{2} c +B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{2} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{2} c}{f}\) | \(186\) |
default | \(\frac {\frac {A \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \cos \left (f x +e \right ) a^{2} c +B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{2} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{2} c}{f}\) | \(186\) |
norman | \(\frac {\left (\frac {1}{2} A \,a^{2} c +\frac {1}{8} B \,a^{2} c \right ) x +\left (2 A \,a^{2} c +\frac {1}{2} B \,a^{2} c \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 A \,a^{2} c +\frac {1}{2} B \,a^{2} c \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 A \,a^{2} c +\frac {3}{4} B \,a^{2} c \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2} c +\frac {1}{8} B \,a^{2} c \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2 A \,a^{2} c +2 B \,a^{2} c}{3 f}-\frac {2 \left (A \,a^{2} c +B \,a^{2} c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {2 \left (A \,a^{2} c +B \,a^{2} c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (A \,a^{2} c +B \,a^{2} c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a^{2} c \left (4 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a^{2} c \left (4 A -B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{2} c \left (4 A +7 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} c \left (4 A +7 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(360\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {8 \, {\left (A + B\right )} a^{2} c \cos \left (f x + e\right )^{3} - 3 \, {\left (4 \, A + B\right )} a^{2} c f x + 3 \, {\left (2 \, B a^{2} c \cos \left (f x + e\right )^{3} - {\left (4 \, A + B\right )} a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (90) = 180\).
Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 4.04 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {A a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {A a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{2} c x + \frac {A a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {A a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 A a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {A a^{2} c \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 B a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {B a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {3 B a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {B a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {5 B a^{2} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {B a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 B a^{2} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {B a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 B a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.83 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c - 96 \, {\left (f x + e\right )} A a^{2} c + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c + 96 \, A a^{2} c \cos \left (f x + e\right ) + 96 \, B a^{2} c \cos \left (f x + e\right )}{96 \, f} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {B a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {A a^{2} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, A a^{2} c + B a^{2} c\right )} x - \frac {{\left (A a^{2} c + B a^{2} c\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (A a^{2} c + B a^{2} c\right )} \cos \left (f x + e\right )}{4 \, f} \]
[In]
[Out]
Time = 13.88 (sec) , antiderivative size = 339, normalized size of antiderivative = 3.46 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,\mathrm {atan}\left (\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A+B\right )}{4\,\left (A\,a^2\,c+\frac {B\,a^2\,c}{4}\right )}\right )\,\left (4\,A+B\right )}{4\,f}-\frac {a^2\,c\,\left (4\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,a^2\,c+2\,B\,a^2\,c\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a^2\,c-\frac {B\,a^2\,c}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a^2\,c+2\,B\,a^2\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a^2\,c}{3}+\frac {2\,B\,a^2\,c}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a^2\,c-\frac {B\,a^2\,c}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a^2\,c+\frac {7\,B\,a^2\,c}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a^2\,c+\frac {7\,B\,a^2\,c}{4}\right )+\frac {2\,A\,a^2\,c}{3}+\frac {2\,B\,a^2\,c}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
[In]
[Out]