Optimal. Leaf size=50 \[ \frac {1}{2} x \left (2 a^2+b^2\right )-\frac {2 a b \cos (e+f x)}{f}-\frac {b^2 \sin (e+f x) \cos (e+f x)}{2 f} \]
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Rubi [A] time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2644} \[ \frac {1}{2} x \left (2 a^2+b^2\right )-\frac {2 a b \cos (e+f x)}{f}-\frac {b^2 \sin (e+f x) \cos (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 2644
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^2 \, dx &=\frac {1}{2} \left (2 a^2+b^2\right ) x-\frac {2 a b \cos (e+f x)}{f}-\frac {b^2 \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 46, normalized size = 0.92 \[ -\frac {-2 \left (2 a^2+b^2\right ) (e+f x)+8 a b \cos (e+f x)+b^2 \sin (2 (e+f x))}{4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 45, normalized size = 0.90 \[ -\frac {b^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (2 \, a^{2} + b^{2}\right )} f x + 4 \, a b \cos \left (f x + e\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 0.90 \[ \frac {1}{2} \, {\left (2 \, a^{2} + b^{2}\right )} x - \frac {2 \, a b \cos \left (f x + e\right )}{f} - \frac {b^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 51, normalized size = 1.02 \[ \frac {b^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-2 a b \cos \left (f x +e \right )+a^{2} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 46, normalized size = 0.92 \[ a^{2} x + \frac {{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2}}{4 \, f} - \frac {2 \, a b \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.79, size = 44, normalized size = 0.88 \[ -\frac {\frac {b^2\,\sin \left (2\,e+2\,f\,x\right )}{2}+4\,a\,b\,\cos \left (e+f\,x\right )-2\,a^2\,f\,x-b^2\,f\,x}{2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 78, normalized size = 1.56 \[ \begin {cases} a^{2} x - \frac {2 a b \cos {\left (e + f x \right )}}{f} + \frac {b^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\relax (e )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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