Optimal. Leaf size=72 \[ -\frac{a (A+7 B) \cos (e+f x)}{3 c^2 f (1-\sin (e+f x))}+\frac{2 a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^2}+\frac{a B x}{c^2} \]
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Rubi [A] time = 0.224578, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2967, 2857, 2735, 2648} \[ -\frac{a (A+7 B) \cos (e+f x)}{3 c^2 f (1-\sin (e+f x))}+\frac{2 a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^2}+\frac{a B x}{c^2} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2857
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx &=(a c) \int \frac{\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx\\ &=\frac{2 a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^2}+\frac{a \int \frac{-A c-4 B c-3 B c \sin (e+f x)}{c-c \sin (e+f x)} \, dx}{3 c^2}\\ &=\frac{a B x}{c^2}+\frac{2 a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^2}-\frac{(a (A+7 B)) \int \frac{1}{c-c \sin (e+f x)} \, dx}{3 c}\\ &=\frac{a B x}{c^2}+\frac{2 a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^2}-\frac{a (A+7 B) \cos (e+f x)}{3 f \left (c^2-c^2 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 0.607064, size = 160, normalized size = 2.22 \[ -\frac{a \left (-6 (A+3 B) \cos \left (e+\frac{f x}{2}\right )+2 A \cos \left (e+\frac{3 f x}{2}\right )+9 B f x \sin \left (e+\frac{f x}{2}\right )+3 B f x \sin \left (e+\frac{3 f x}{2}\right )+14 B \cos \left (e+\frac{3 f x}{2}\right )+3 B f x \cos \left (2 e+\frac{3 f x}{2}\right )+24 B \sin \left (\frac{f x}{2}\right )-9 B f x \cos \left (\frac{f x}{2}\right )\right )}{6 c^2 f \left (\cos \left (\frac{e}{2}\right )-\sin \left (\frac{e}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 160, normalized size = 2.2 \begin{align*} -2\,{\frac{aA}{f{c}^{2} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}+2\,{\frac{Ba}{f{c}^{2} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-{\frac{8\,aA}{3\,f{c}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-3}}-{\frac{8\,Ba}{3\,f{c}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-3}}-4\,{\frac{aA}{f{c}^{2} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-4\,{\frac{Ba}{f{c}^{2} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}+2\,{\frac{Ba\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48382, size = 616, normalized size = 8.56 \begin{align*} \frac{2 \,{\left (B a{\left (\frac{\frac{9 \, \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}} - 4}{c^{2} - \frac{3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac{3 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{2}}\right )} - \frac{A a{\left (\frac{3 \, \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}} - 2\right )}}{c^{2} - \frac{3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac{A a{\left (\frac{3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}}{c^{2} - \frac{3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac{B a{\left (\frac{3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}}{c^{2} - \frac{3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42143, size = 402, normalized size = 5.58 \begin{align*} -\frac{6 \, B a f x -{\left (3 \, B a f x +{\left (A + 7 \, B\right )} a\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (A + B\right )} a +{\left (3 \, B a f x +{\left (A - 5 \, B\right )} a\right )} \cos \left (f x + e\right ) -{\left (6 \, B a f x - 2 \,{\left (A + B\right )} a +{\left (3 \, B a f x -{\left (A + 7 \, B\right )} a\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \,{\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f +{\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \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 = 19.9417, size = 711, normalized size = 9.88 \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 [A] time = 1.14274, size = 124, normalized size = 1.72 \begin{align*} \frac{\frac{3 \,{\left (f x + e\right )} B a}{c^{2}} - \frac{2 \,{\left (3 \, A a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 3 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 12 \, B a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + A a - 5 \, B a\right )}}{c^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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