Optimal. Leaf size=52 \[ \frac{2 \tan (e+f x)}{3 a^2 c f}-\frac{\sec (e+f x)}{3 c f \left (a^2 \sin (e+f x)+a^2\right )} \]
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Rubi [A] time = 0.10668, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2736, 2672, 3767, 8} \[ \frac{2 \tan (e+f x)}{3 a^2 c f}-\frac{\sec (e+f x)}{3 c f \left (a^2 \sin (e+f x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx &=\frac{\int \frac{\sec ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a c}\\ &=-\frac{\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{2 \int \sec ^2(e+f x) \, dx}{3 a^2 c}\\ &=-\frac{\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a^2 c f}\\ &=-\frac{\sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac{2 \tan (e+f x)}{3 a^2 c f}\\ \end{align*}
Mathematica [A] time = 0.461606, size = 87, normalized size = 1.67 \[ -\frac{\sin (e+f x)+8 \sin (2 (e+f x))+\sin (3 (e+f x))-4 \cos (e+f x)+2 \cos (2 (e+f x))-4 \cos (3 (e+f x))+2}{24 a^2 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 73, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{{a}^{2}cf} \left ( -1/4\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-1/3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-3}+1/2\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-2}-3/4\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55873, size = 192, normalized size = 3.69 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - 1\right )}}{3 \,{\left (a^{2} c + \frac{2 \, a^{2} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{2 \, a^{2} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac{a^{2} c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37038, size = 142, normalized size = 2.73 \begin{align*} -\frac{2 \, \cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 1}{3 \,{\left (a^{2} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} c f \cos \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.1402, size = 328, normalized size = 6.31 \begin{align*} \begin{cases} - \frac{\tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{3 a^{2} c f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 6 a^{2} c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c f} - \frac{8 \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{3 a^{2} c f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 6 a^{2} c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c f} - \frac{6 \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{3 a^{2} c f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 6 a^{2} c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c f} + \frac{3}{3 a^{2} c f \tan ^{4}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 6 a^{2} c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 6 a^{2} c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - 3 a^{2} c f} & \text{for}\: f \neq 0 \\\frac{x}{\left (a \sin{\left (e \right )} + a\right )^{2} \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 = 1.98881, size = 104, normalized size = 2. \begin{align*} -\frac{\frac{3}{a^{2} c{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}} + \frac{9 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 12 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 7}{a^{2} c{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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