Integrand size = 24, antiderivative size = 52 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {1}{2} a c^2 x+\frac {a c^2 \cos ^3(e+f x)}{3 f}+\frac {a c^2 \cos (e+f x) \sin (e+f x)}{2 f} \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2815, 2748, 2715, 8} \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {a c^2 \cos ^3(e+f x)}{3 f}+\frac {a c^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} a c^2 x \]
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Rule 8
Rule 2715
Rule 2748
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {a c^2 \cos ^3(e+f x)}{3 f}+\left (a c^2\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {a c^2 \cos ^3(e+f x)}{3 f}+\frac {a c^2 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} \left (a c^2\right ) \int 1 \, dx \\ & = \frac {1}{2} a c^2 x+\frac {a c^2 \cos ^3(e+f x)}{3 f}+\frac {a c^2 \cos (e+f x) \sin (e+f x)}{2 f} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {a c^2 (6 f x+3 \cos (e+f x)+\cos (3 (e+f x))+3 \sin (2 (e+f x)))}{12 f} \]
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Time = 1.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {a \,c^{2} \left (6 f x +\cos \left (3 f x +3 e \right )+3 \sin \left (2 f x +2 e \right )+3 \cos \left (f x +e \right )+4\right )}{12 f}\) | \(44\) |
risch | \(\frac {a \,c^{2} x}{2}+\frac {a \,c^{2} \cos \left (f x +e \right )}{4 f}+\frac {a \,c^{2} \cos \left (3 f x +3 e \right )}{12 f}+\frac {a \,c^{2} \sin \left (2 f x +2 e \right )}{4 f}\) | \(60\) |
derivativedivides | \(\frac {-\frac {a \,c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-a \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a \,c^{2} \cos \left (f x +e \right )+a \,c^{2} \left (f x +e \right )}{f}\) | \(77\) |
default | \(\frac {-\frac {a \,c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-a \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a \,c^{2} \cos \left (f x +e \right )+a \,c^{2} \left (f x +e \right )}{f}\) | \(77\) |
parts | \(a \,c^{2} x -\frac {a \,c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {a \,c^{2} \cos \left (f x +e \right )}{f}-\frac {a \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(78\) |
norman | \(\frac {\frac {2 a \,c^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a \,c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {2 a \,c^{2}}{3 f}+\frac {a \,c^{2} x}{2}-\frac {a \,c^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a \,c^{2} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {3 a \,c^{2} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {a \,c^{2} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\) | \(145\) |
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {2 \, a c^{2} \cos \left (f x + e\right )^{3} + 3 \, a c^{2} f x + 3 \, a c^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{6 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (46) = 92\).
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.56 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\begin {cases} - \frac {a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a c^{2} x - \frac {a c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {a c^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )^{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.48 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a c^{2} - 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a c^{2} + 12 \, {\left (f x + e\right )} a c^{2} + 12 \, a c^{2} \cos \left (f x + e\right )}{12 \, f} \]
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Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {1}{2} \, a c^{2} x + \frac {a c^{2} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {a c^{2} \cos \left (f x + e\right )}{4 \, f} + \frac {a c^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 9.66 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.40 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx=\frac {a\,c^2\,x}{2}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {3\,a\,c^2\,\left (e+f\,x\right )}{2}-\frac {a\,c^2\,\left (9\,e+9\,f\,x+12\right )}{6}\right )-a\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {a\,c^2\,\left (e+f\,x\right )}{2}+a\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {a\,c^2\,\left (3\,e+3\,f\,x+4\right )}{6}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^3} \]
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