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 = 0.0161034, antiderivative size = 50, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0970928, 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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Maple [A] time = 0.018, size = 51, normalized size = 1. \begin{align*}{\frac{1}{f} \left ({b}^{2} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -2\,ab\cos \left ( fx+e \right ) +{a}^{2} \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01385, size = 62, normalized size = 1.24 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53957, size = 109, normalized size = 2.18 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.308759, size = 78, normalized size = 1.56 \begin{align*} \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{\left (e \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24285, size = 61, normalized size = 1.22 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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