Optimal. Leaf size=16 \[ \frac{\sin (e+f x) \cos (e+f x)}{f} \]
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Rubi [A] time = 0.0126719, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2635, 8} \[ \frac{\sin (e+f x) \cos (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (1-2 \sin ^2(e+f x)\right ) \, dx &=x-2 \int \sin ^2(e+f x) \, dx\\ &=x+\frac{\cos (e+f x) \sin (e+f x)}{f}-\int 1 \, dx\\ &=\frac{\cos (e+f x) \sin (e+f x)}{f}\\ \end{align*}
Mathematica [B] time = 0.0080478, size = 33, normalized size = 2.06 \[ \frac{\sin (2 e) \cos (2 f x)}{2 f}+\frac{\cos (2 e) \sin (2 f x)}{2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 30, normalized size = 1.9 \begin{align*} x-2\,{\frac{-1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939203, size = 35, normalized size = 2.19 \begin{align*} x - \frac{2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63091, size = 39, normalized size = 2.44 \begin{align*} \frac{\cos \left (f x + e\right ) \sin \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.370381, size = 49, normalized size = 3.06 \begin{align*} x - 2 \left (\begin{cases} \frac{x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{\sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} & \text{for}\: f \neq 0 \\x \sin ^{2}{\left (e \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11331, size = 20, normalized size = 1.25 \begin{align*} \frac{\sin \left (2 \, f x + 2 \, e\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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