Optimal. Leaf size=35 \[ \frac {d x}{a}-\frac {(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)} \]
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Rubi [A] time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2735, 2648} \[ \frac {d x}{a}-\frac {(c-d) \cos (e+f x)}{f (a \sin (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {c+d \sin (e+f x)}{a+a \sin (e+f x)} \, dx &=\frac {d x}{a}-(-c+d) \int \frac {1}{a+a \sin (e+f x)} \, dx\\ &=\frac {d x}{a}-\frac {(c-d) \cos (e+f x)}{f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [B] time = 0.17, size = 79, normalized size = 2.26 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right ) (2 c+d (e+f x-2))+d (e+f x) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{a f (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 66, normalized size = 1.89 \[ \frac {d f x + {\left (d f x - c + d\right )} \cos \left (f x + e\right ) + {\left (d f x + c - d\right )} \sin \left (f x + e\right ) - c + d}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 40, normalized size = 1.14 \[ \frac {\frac {{\left (f x + e\right )} d}{a} - \frac {2 \, {\left (c - d\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 65, normalized size = 1.86 \[ \frac {2 d \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2 c}{a f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {2 d}{a f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 78, normalized size = 2.23 \[ \frac {2 \, {\left (d {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {c}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.83, size = 35, normalized size = 1.00 \[ \frac {d\,x}{a}-\frac {2\,c-2\,d}{a\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 109, normalized size = 3.11 \[ \begin {cases} - \frac {2 c}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {d f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {d f x}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2 d}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} & \text {for}\: f \neq 0 \\\frac {x \left (c + d \sin {\relax (e )}\right )}{a \sin {\relax (e )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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