Optimal. Leaf size=57 \[ \frac{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac{3 a^2 x}{c} \]
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Rubi [A] time = 0.140906, antiderivative size = 57, 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, 2680, 2682, 8} \[ \frac{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac{3 a^2 x}{c} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2680
Rule 2682
Rule 8
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^2}{c-c \sin (e+f x)} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^3} \, dx\\ &=\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\left (3 a^2\right ) \int \frac{\cos ^2(e+f x)}{c-c \sin (e+f x)} \, dx\\ &=\frac{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac{\left (3 a^2\right ) \int 1 \, dx}{c}\\ &=-\frac{3 a^2 x}{c}+\frac{3 a^2 \cos (e+f x)}{c f}+\frac{2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}\\ \end{align*}
Mathematica [B] time = 0.371187, size = 130, normalized size = 2.28 \[ \frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right ) (3 (e+f x)-\cos (e+f x))+\sin \left (\frac{1}{2} (e+f x)\right ) (\cos (e+f x)-3 e-3 f x-8)\right )}{c f (\sin (e+f x)-1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 73, normalized size = 1.3 \begin{align*} -8\,{\frac{{a}^{2}}{cf \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}+2\,{\frac{{a}^{2}}{cf \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) }}-6\,{\frac{{a}^{2}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{cf}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.70054, size = 294, normalized size = 5.16 \begin{align*} -\frac{2 \,{\left (a^{2}{\left (\frac{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 2}{c - \frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac{\arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c}\right )} + 2 \, a^{2}{\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^{2}}{c - \frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32484, size = 238, normalized size = 4.18 \begin{align*} -\frac{3 \, a^{2} f x - a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{2} +{\left (3 \, a^{2} f x - 5 \, a^{2}\right )} \cos \left (f x + e\right ) -{\left (3 \, a^{2} f x - a^{2} \cos \left (f x + e\right ) + 4 \, a^{2}\right )} \sin \left (f x + e\right )}{c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.49521, size = 456, normalized size = 8. \begin{align*} \begin{cases} - \frac{3 a^{2} f x \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} + \frac{3 a^{2} f x \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} - \frac{3 a^{2} f x \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} + \frac{3 a^{2} f x}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} - \frac{2 a^{2} \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} - \frac{6 a^{2} \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} - \frac{8 a^{2}}{c f \tan ^{3}{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f \tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + c f \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} - c f} & \text{for}\: f \neq 0 \\\frac{x \left (a \sin{\left (e \right )} + a\right )^{2}}{- c \sin{\left (e \right )} + c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.06178, size = 139, normalized size = 2.44 \begin{align*} -\frac{\frac{3 \,{\left (f x + e\right )} a^{2}}{c} + \frac{2 \,{\left (4 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 5 \, a^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )} c}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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