Optimal. Leaf size=32 \[ -\frac {2 c \cos (e+f x)}{f (a \sin (e+f x)+a)}-\frac {c x}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2735, 2648} \[ -\frac {2 c \cos (e+f x)}{f (a \sin (e+f x)+a)}-\frac {c x}{a} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {c-c \sin (e+f x)}{a+a \sin (e+f x)} \, dx &=-\frac {c x}{a}+(2 c) \int \frac {1}{a+a \sin (e+f x)} \, dx\\ &=-\frac {c x}{a}-\frac {2 c \cos (e+f x)}{f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 79, normalized size = 2.47 \[ -\frac {c \left (f x \sin \left (e+\frac {f x}{2}\right )-4 \sin \left (\frac {f x}{2}\right )+f x \cos \left (\frac {f x}{2}\right )\right )}{a f \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 64, normalized size = 2.00 \[ -\frac {c f x + {\left (c f x + 2 \, c\right )} \cos \left (f x + e\right ) + {\left (c f x - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 37, normalized size = 1.16 \[ -\frac {\frac {{\left (f x + e\right )} c}{a} + \frac {4 \, c}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 43, normalized size = 1.34 \[ -\frac {2 c \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {4 c}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.86, size = 77, normalized size = 2.41 \[ -\frac {2 \, {\left (c {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + \frac {c}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.64, size = 45, normalized size = 1.41 \[ \frac {c\,\left (e+f\,x\right )-c\,\left (e+f\,x+4\right )}{a\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}-\frac {c\,x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 90, normalized size = 2.81 \[ \begin {cases} - \frac {c f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {c f x}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {4 c}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} & \text {for}\: f \neq 0 \\\frac {x \left (- c \sin {\relax (e )} + c\right )}{a \sin {\relax (e )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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