Optimal. Leaf size=56 \[ \frac{2 a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{a x (A+2 B)}{c}+\frac{a B \cos (e+f x)}{c f} \]
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Rubi [A] time = 0.169462, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {2967, 2857, 2638} \[ \frac{2 a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{a x (A+2 B)}{c}+\frac{a B \cos (e+f x)}{c f} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2857
Rule 2638
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx &=(a c) \int \frac{\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx\\ &=\frac{2 a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))}+\frac{a \int (-A c-2 B c-B c \sin (e+f x)) \, dx}{c^2}\\ &=-\frac{a (A+2 B) x}{c}+\frac{2 a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))}-\frac{(a B) \int \sin (e+f x) \, dx}{c}\\ &=-\frac{a (A+2 B) x}{c}+\frac{a B \cos (e+f x)}{c f}+\frac{2 a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))}\\ \end{align*}
Mathematica [B] time = 0.862014, size = 125, normalized size = 2.23 \[ \frac{a (\sin (e+f x)+1) \left (\frac{4 (A+B) \sin \left (\frac{f x}{2}\right )}{f \left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}+x (-(A+2 B))-\frac{B \sin (e) \sin (f x)}{f}+\frac{B \cos (e) \cos (f x)}{f}\right )}{c \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.1, size = 113, normalized size = 2. \begin{align*} -4\,{\frac{aA}{cf \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-4\,{\frac{Ba}{cf \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}+2\,{\frac{Ba}{cf \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{a\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) A}{cf}}-4\,{\frac{a\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) B}{cf}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4559, size = 358, normalized size = 6.39 \begin{align*} -\frac{2 \,{\left (B a{\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 )} + A 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 )} + B 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 a}{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 [B] time = 1.40548, size = 289, normalized size = 5.16 \begin{align*} -\frac{{\left (A + 2 \, B\right )} a f x - B a \cos \left (f x + e\right )^{2} - 2 \,{\left (A + B\right )} a +{\left ({\left (A + 2 \, B\right )} a f x -{\left (2 \, A + 3 \, B\right )} a\right )} \cos \left (f x + e\right ) -{\left ({\left (A + 2 \, B\right )} a f x - B a \cos \left (f x + e\right ) + 2 \,{\left (A + B\right )} a\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 = 10.4835, size = 830, normalized size = 14.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19904, size = 167, normalized size = 2.98 \begin{align*} -\frac{\frac{{\left (A a + 2 \, B a\right )}{\left (f x + e\right )}}{c} + \frac{2 \,{\left (2 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2 \, A a + 3 \, B a\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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