Optimal. Leaf size=48 \[ -\frac {a (A+B) \cos (e+f x)}{f}+\frac {1}{2} a x (2 A+B)-\frac {a B \sin (e+f x) \cos (e+f x)}{2 f} \]
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Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2734} \[ -\frac {a (A+B) \cos (e+f x)}{f}+\frac {1}{2} a x (2 A+B)-\frac {a B \sin (e+f x) \cos (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 2734
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx &=\frac {1}{2} a (2 A+B) x-\frac {a (A+B) \cos (e+f x)}{f}-\frac {a B \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 45, normalized size = 0.94 \[ \frac {a (-4 (A+B) \cos (e+f x)+4 A f x-B \sin (2 (e+f x))+2 B e+2 B f x)}{4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 43, normalized size = 0.90 \[ \frac {{\left (2 \, A + B\right )} a f x - B a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, {\left (A + B\right )} a \cos \left (f x + e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 48, normalized size = 1.00 \[ \frac {1}{2} \, {\left (2 \, A a + B a\right )} x - \frac {B a \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} - \frac {{\left (A a + B a\right )} \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 59, normalized size = 1.23 \[ \frac {a B \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a A \cos \left (f x +e \right )-a B \cos \left (f x +e \right )+a A \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 57, normalized size = 1.19 \[ \frac {4 \, {\left (f x + e\right )} A a + {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a - 4 \, A a \cos \left (f x + e\right ) - 4 \, B a \cos \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.26, size = 100, normalized size = 2.08 \[ A\,a\,x-\frac {-B\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+B\,a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+2\,A\,a+2\,B\,a}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {B\,a\,x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 94, normalized size = 1.96 \[ \begin {cases} A a x - \frac {A a \cos {\left (e + f x \right )}}{f} + \frac {B a x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {B a x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {B a \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {B a \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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