Optimal. Leaf size=16 \[ a x-\frac {a \cos (e+f x)}{f} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2638} \[ a x-\frac {a \cos (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 2638
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) \, dx &=a x+a \int \sin (e+f x) \, dx\\ &=a x-\frac {a \cos (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.69 \[ \frac {a \sin (e) \sin (f x)}{f}-\frac {a \cos (e) \cos (f x)}{f}+a x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 18, normalized size = 1.12 \[ \frac {a f x - a \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 17, normalized size = 1.06 \[ a x - \frac {a \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 17, normalized size = 1.06 \[ a x -\frac {a \cos \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 16, normalized size = 1.00 \[ a x - \frac {a \cos \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.72, size = 25, normalized size = 1.56 \[ a\,x-\frac {2\,a}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 19, normalized size = 1.19 \[ a x + a \left (\begin {cases} - \frac {\cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \sin {\relax (e )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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