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 = 0.17, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 2638
Rule 2857
Rule 2967
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*}
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Mathematica [B] time = 0.93, 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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fricas [B] time = 0.42, size = 116, normalized size = 2.07 \[ -\frac {{\left (A + 2 \, B\right )} a f x - B a \cos \left (f x + e\right )^{2} - 2 \, {\left (A + B\right )} a + {\left ({\left (A + 2 \, B\right )} a f x - {\left (2 \, A + 3 \, B\right )} a\right )} \cos \left (f x + e\right ) - {\left ({\left (A + 2 \, B\right )} a f x - B a \cos \left (f x + e\right ) + 2 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )}{c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 124, normalized size = 2.21 \[ -\frac {\frac {{\left (A a + 2 \, B a\right )} {\left (f x + e\right )}}{c} + \frac {2 \, {\left (2 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A a + 3 \, B a\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} c}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 113, normalized size = 2.02 \[ -\frac {4 a A}{f c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {4 a B}{f c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {2 a B}{f c \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}-\frac {2 a \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) A}{f c}-\frac {4 a \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) B}{f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 265, normalized size = 4.73 \[ -\frac {2 \, {\left (B a {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 2}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c}\right )} + A a {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} - \frac {1}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + B a {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} - \frac {1}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {A a}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.64, size = 111, normalized size = 1.98 \[ \frac {\left (4\,A\,a+4\,B\,a\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-2\,B\,a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+4\,A\,a+6\,B\,a}{f\,\left (-c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+c\right )}-\frac {A\,a\,f\,x+2\,B\,a\,f\,x}{c\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.03, size = 828, normalized size = 14.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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