Optimal. Leaf size=35 \[ \frac{A \tan (e+f x)}{a c f}+\frac{B \sec (e+f x)}{a c f} \]
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Rubi [A] time = 0.135558, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2669, 3767, 8} \[ \frac{A \tan (e+f x)}{a c f}+\frac{B \sec (e+f x)}{a c f} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx &=\frac{\int \sec ^2(e+f x) (A+B \sin (e+f x)) \, dx}{a c}\\ &=\frac{B \sec (e+f x)}{a c f}+\frac{A \int \sec ^2(e+f x) \, dx}{a c}\\ &=\frac{B \sec (e+f x)}{a c f}-\frac{A \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a c f}\\ &=\frac{B \sec (e+f x)}{a c f}+\frac{A \tan (e+f x)}{a c f}\\ \end{align*}
Mathematica [A] time = 0.0283891, size = 35, normalized size = 1. \[ \frac{A \tan (e+f x)}{a c f}+\frac{B \sec (e+f x)}{a c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 57, normalized size = 1.6 \begin{align*} 2\,{\frac{1}{acf} \left ( -{\frac{A/2-B/2}{\tan \left ( 1/2\,fx+e/2 \right ) +1}}-{\frac{A/2+B/2}{\tan \left ( 1/2\,fx+e/2 \right ) -1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9652, size = 47, normalized size = 1.34 \begin{align*} \frac{\frac{A \tan \left (f x + e\right )}{a c} + \frac{B}{a c \cos \left (f x + e\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35681, size = 58, normalized size = 1.66 \begin{align*} \frac{A \sin \left (f x + e\right ) + B}{a c f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.16627, size = 83, normalized size = 2.37 \begin{align*} \begin{cases} - \frac{2 A \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{a c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - a c f} - \frac{2 B}{a c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - a c f} & \text{for}\: f \neq 0 \\\frac{x \left (A + B \sin{\left (e \right )}\right )}{\left (a \sin{\left (e \right )} + a\right ) \left (- c \sin{\left (e \right )} + c\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20056, size = 55, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + B\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )} a c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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