Optimal. Leaf size=52 \[ \frac {a c^2 \cos ^3(e+f x)}{3 f}+\frac {a c^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} a c^2 x \]
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Rubi [A] time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2736, 2669, 2635, 8} \[ \frac {a c^2 \cos ^3(e+f x)}{3 f}+\frac {a c^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} a c^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2736
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx &=(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*}
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Mathematica [A] time = 0.28, size = 42, normalized size = 0.81 \[ \frac {a c^2 (3 \sin (2 (e+f x))+3 \cos (e+f x)+\cos (3 (e+f x))+6 f x)}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 46, normalized size = 0.88 \[ \frac {2 \, a c^{2} \cos \left (f x + e\right )^{3} + 3 \, a c^{2} f x + 3 \, a c^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 62, normalized size = 1.19 \[ \frac {1}{2} \, a c^{2} x + \frac {a c^{2} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {a c^{2} \cos \left (f x + e\right )}{4 \, f} + \frac {a c^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 77, normalized size = 1.48 \[ \frac {-\frac {a \,c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-a \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a \,c^{2} \cos \left (f x +e \right )+a \,c^{2} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 77, normalized size = 1.48 \[ \frac {4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a c^{2} - 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a c^{2} + 12 \, {\left (f x + e\right )} a c^{2} + 12 \, a c^{2} \cos \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.96, size = 125, normalized size = 2.40 \[ \frac {a\,c^2\,x}{2}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {3\,a\,c^2\,\left (e+f\,x\right )}{2}-\frac {a\,c^2\,\left (9\,e+9\,f\,x+12\right )}{6}\right )-a\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {a\,c^2\,\left (e+f\,x\right )}{2}+a\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {a\,c^2\,\left (3\,e+3\,f\,x+4\right )}{6}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 133, normalized size = 2.56 \[ \begin {cases} - \frac {a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a c^{2} x - \frac {a c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {a c^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\relax (e )} + a\right ) \left (- c \sin {\relax (e )} + c\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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