Integrand size = 34, antiderivative size = 97 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {1}{8} a (4 A-B) c^2 x+\frac {a (A-B) c^2 \cos ^3(e+f x)}{3 f}+\frac {a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a B c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f} \]
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Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08, 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} \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {a c^2 (4 A-B) \cos ^3(e+f x)}{12 f}+\frac {a c^2 (4 A-B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a c^2 x (4 A-B)-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f} \]
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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+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx \\ & = -\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} (a (4 A-B) c) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} \left (a (4 A-B) c^2\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}+\frac {a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac {1}{8} \left (a (4 A-B) c^2\right ) \int 1 \, dx \\ & = \frac {1}{8} a (4 A-B) c^2 x+\frac {a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}+\frac {a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {a c^2 \cos (e+f x) \left (8 A-8 B-\frac {6 (4 A-B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}+3 (4 A+B) \sin (e+f x)-8 (A-B) \sin ^2(e+f x)-6 B \sin ^3(e+f x)\right )}{24 f} \]
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Time = 1.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {c^{2} \left (\frac {\left (A -B \right ) \cos \left (3 f x +3 e \right )}{3}+A \sin \left (2 f x +2 e \right )+\frac {B \sin \left (4 f x +4 e \right )}{8}+\left (A -B \right ) \cos \left (f x +e \right )+2 f x A -\frac {f x B}{2}+\frac {4 A}{3}-\frac {4 B}{3}\right ) a}{4 f}\) | \(78\) |
risch | \(\frac {a \,c^{2} x A}{2}-\frac {a \,c^{2} x B}{8}+\frac {c^{2} a \cos \left (f x +e \right ) A}{4 f}-\frac {c^{2} a \cos \left (f x +e \right ) B}{4 f}+\frac {B \,c^{2} a \sin \left (4 f x +4 e \right )}{32 f}+\frac {c^{2} a \cos \left (3 f x +3 e \right ) A}{12 f}-\frac {c^{2} a \cos \left (3 f x +3 e \right ) B}{12 f}+\frac {A \,c^{2} a \sin \left (2 f x +2 e \right )}{4 f}\) | \(126\) |
parts | \(\frac {\left (-A \,c^{2} a -B \,c^{2} a \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (-A \,c^{2} a +B \,c^{2} a \right ) \cos \left (f x +e \right )}{f}-\frac {\left (A \,c^{2} a -B \,c^{2} a \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+a \,c^{2} x A +\frac {B \,c^{2} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(152\) |
derivativedivides | \(\frac {-\frac {A \,c^{2} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,c^{2} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \cos \left (f x +e \right ) a \,c^{2}+B \,c^{2} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,c^{2} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,c^{2} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{2} a \left (f x +e \right )-B \cos \left (f x +e \right ) a \,c^{2}}{f}\) | \(185\) |
default | \(\frac {-\frac {A \,c^{2} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,c^{2} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \cos \left (f x +e \right ) a \,c^{2}+B \,c^{2} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,c^{2} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,c^{2} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{2} a \left (f x +e \right )-B \cos \left (f x +e \right ) a \,c^{2}}{f}\) | \(185\) |
norman | \(\frac {\left (\frac {1}{2} A \,c^{2} a -\frac {1}{8} B \,c^{2} a \right ) x +\left (2 A \,c^{2} a -\frac {1}{2} B \,c^{2} a \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 A \,c^{2} a -\frac {1}{2} B \,c^{2} a \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 A \,c^{2} a -\frac {3}{4} B \,c^{2} a \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {1}{2} A \,c^{2} a -\frac {1}{8} B \,c^{2} a \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 A \,c^{2} a -2 B \,c^{2} a}{3 f}+\frac {2 \left (A \,c^{2} a -B \,c^{2} a \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (A \,c^{2} a -B \,c^{2} a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (A \,c^{2} a -B \,c^{2} a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {a \,c^{2} \left (4 A -7 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a \,c^{2} \left (4 A -7 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a \,c^{2} \left (4 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a \,c^{2} \left (4 A +B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(359\) |
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Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {8 \, {\left (A - B\right )} a c^{2} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, A - B\right )} a c^{2} f x + 3 \, {\left (2 \, B a c^{2} \cos \left (f x + e\right )^{3} + {\left (4 \, A - B\right )} a c^{2} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (85) = 170\).
Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 4.08 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\begin {cases} - \frac {A a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {A a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + A a c^{2} x - \frac {A a c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {A a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {A a c^{2} \cos {\left (e + f x \right )}}{f} + \frac {3 B a c^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {B a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 B a c^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {B a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {5 B a c^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {B a c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 B a c^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {B a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 B a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a c^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )^{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (89) = 178\).
Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.85 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a c^{2} - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{2} + 96 \, {\left (f x + e\right )} A a c^{2} - 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c^{2} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} + 96 \, A a c^{2} \cos \left (f x + e\right ) - 96 \, B a c^{2} \cos \left (f x + e\right )}{96 \, f} \]
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Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {B a c^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {A a c^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, A a c^{2} - B a c^{2}\right )} x + \frac {{\left (A a c^{2} - B a c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {{\left (A a c^{2} - B a c^{2}\right )} \cos \left (f x + e\right )}{4 \, f} \]
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Time = 13.87 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.56 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a\,c^2+\frac {B\,a\,c^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,a\,c^2-2\,B\,a\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a\,c^2-2\,B\,a\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a\,c^2}{3}-\frac {2\,B\,a\,c^2}{3}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a\,c^2+\frac {B\,a\,c^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a\,c^2-\frac {7\,B\,a\,c^2}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a\,c^2-\frac {7\,B\,a\,c^2}{4}\right )+\frac {2\,A\,a\,c^2}{3}-\frac {2\,B\,a\,c^2}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a\,c^2\,\mathrm {atan}\left (\frac {a\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A-B\right )}{4\,\left (A\,a\,c^2-\frac {B\,a\,c^2}{4}\right )}\right )\,\left (4\,A-B\right )}{4\,f}-\frac {a\,c^2\,\left (4\,A-B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]
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