Optimal. Leaf size=33 \[ \frac {2 a \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac {a x}{c} \]
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Rubi [A] time = 0.05, antiderivative size = 33, 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} \[ \frac {2 a \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac {a x}{c} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{c-c \sin (e+f x)} \, dx &=-\frac {a x}{c}+(2 a) \int \frac {1}{c-c \sin (e+f x)} \, dx\\ &=-\frac {a x}{c}+\frac {2 a \cos (e+f x)}{f (c-c \sin (e+f x))}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 83, normalized size = 2.52 \[ \frac {a \left (f x \sin \left (e+\frac {f x}{2}\right )+4 \sin \left (\frac {f x}{2}\right )-f x \cos \left (\frac {f x}{2}\right )\right )}{c f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 66, normalized size = 2.00 \[ -\frac {a f x + {\left (a f x - 2 \, a\right )} \cos \left (f x + e\right ) - {\left (a f x + 2 \, a\right )} \sin \left (f x + e\right ) - 2 \, a}{c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 37, normalized size = 1.12 \[ -\frac {\frac {{\left (f x + e\right )} a}{c} + \frac {4 \, a}{c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 43, normalized size = 1.30 \[ -\frac {4 a}{f c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {2 a \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.92, size = 82, normalized size = 2.48 \[ -\frac {2 \, {\left (a {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} - \frac {1}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {a}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.81, size = 46, normalized size = 1.39 \[ -\frac {a\,x}{c}-\frac {a\,\left (e+f\,x\right )-a\,\left (e+f\,x-4\right )}{c\,f\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.61, size = 88, normalized size = 2.67 \[ \begin {cases} - \frac {a f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} + \frac {a f x}{c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} - \frac {4 a}{c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\relax (e )} + a\right )}{- c \sin {\relax (e )} + c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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