Optimal. Leaf size=94 \[ -\frac {2 a^2 (3 A+2 B) \cos (e+f x)}{3 f}-\frac {a^2 (3 A+2 B) \sin (e+f x) \cos (e+f x)}{6 f}+\frac {1}{2} a^2 x (3 A+2 B)-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f} \]
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Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2751, 2644} \[ -\frac {2 a^2 (3 A+2 B) \cos (e+f x)}{3 f}-\frac {a^2 (3 A+2 B) \sin (e+f x) \cos (e+f x)}{6 f}+\frac {1}{2} a^2 x (3 A+2 B)-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f} \]
Antiderivative was successfully verified.
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Rule 2644
Rule 2751
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx &=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^2}{3 f}+\frac {1}{3} (3 A+2 B) \int (a+a \sin (e+f x))^2 \, dx\\ &=\frac {1}{2} a^2 (3 A+2 B) x-\frac {2 a^2 (3 A+2 B) \cos (e+f x)}{3 f}-\frac {a^2 (3 A+2 B) \cos (e+f x) \sin (e+f x)}{6 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^2}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 106, normalized size = 1.13 \[ -\frac {a^2 \cos (e+f x) \left (6 (3 A+2 B) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (3 (A+2 B) \sin (e+f x)+2 (6 A+5 B)+2 B \sin ^2(e+f x)\right )\right )}{6 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 70, normalized size = 0.74 \[ \frac {2 \, B a^{2} \cos \left (f x + e\right )^{3} + 3 \, {\left (3 \, A + 2 \, B\right )} a^{2} f x - 3 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 12 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 0.94 \[ \frac {B a^{2} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {1}{2} \, {\left (3 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac {{\left (8 \, A a^{2} + 7 \, B a^{2}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 117, normalized size = 1.24 \[ \frac {a^{2} A \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {B \,a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 a^{2} A \cos \left (f x +e \right )+2 B \,a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{2} A \left (f x +e \right )-B \,a^{2} \cos \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 114, normalized size = 1.21 \[ \frac {3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} + 12 \, {\left (f x + e\right )} A a^{2} + 4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} + 6 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} - 24 \, A a^{2} \cos \left (f x + e\right ) - 12 \, B a^{2} \cos \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.16, size = 91, normalized size = 0.97 \[ -\frac {\frac {3\,A\,a^2\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {B\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{2}+3\,B\,a^2\,\sin \left (2\,e+2\,f\,x\right )+12\,A\,a^2\,\cos \left (e+f\,x\right )+\frac {21\,B\,a^2\,\cos \left (e+f\,x\right )}{2}-9\,A\,a^2\,f\,x-6\,B\,a^2\,f\,x}{6\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 199, normalized size = 2.12 \[ \begin {cases} \frac {A a^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {A a^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{2} x - \frac {A a^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a^{2} \cos {\left (e + f x \right )}}{f} + B a^{2} x \sin ^{2}{\left (e + f x \right )} + B a^{2} x \cos ^{2}{\left (e + f x \right )} - \frac {B a^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {B a^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 B a^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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