Optimal. Leaf size=62 \[ \frac {(2 A+B) \tan (e+f x)}{3 a^2 c f}-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2 \sin (e+f x)+a^2\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2967, 2859, 3767, 8} \[ \frac {(2 A+B) \tan (e+f x)}{3 a^2 c f}-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2 \sin (e+f x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2859
Rule 2967
Rule 3767
Rubi steps
\begin {align*} \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))} \, dx &=\frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{a c}\\ &=-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {(2 A+B) \int \sec ^2(e+f x) \, dx}{3 a^2 c}\\ &=-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}-\frac {(2 A+B) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a^2 c f}\\ &=-\frac {(A-B) \sec (e+f x)}{3 c f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {(2 A+B) \tan (e+f x)}{3 a^2 c f}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 110, normalized size = 1.77 \[ \frac {\cos (e+f x) (-2 (A-B) \cos (e+f x)+2 (2 A+B) \cos (2 (e+f x))-8 A \sin (e+f x)-A \sin (2 (e+f x))-4 B \sin (e+f x)+B \sin (2 (e+f x))-6 B)}{12 a^2 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 69, normalized size = 1.11 \[ -\frac {{\left (2 \, A + B\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, A + B\right )} \sin \left (f x + e\right ) - A - 2 \, B}{3 \, {\left (a^{2} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} c f \cos \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 102, normalized size = 1.65 \[ -\frac {\frac {3 \, {\left (A + B\right )}}{a^{2} c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}} + \frac {9 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, A - B}{a^{2} c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 97, normalized size = 1.56 \[ \frac {-\frac {2 \left (\frac {A}{4}+\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {-A +B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (A -B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {3 A}{4}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{2} c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 265, normalized size = 4.27 \[ \frac {2 \, {\left (\frac {B {\left (\frac {2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{2} c + \frac {2 \, a^{2} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a^{2} c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {A {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - 1\right )}}{a^{2} c + \frac {2 \, a^{2} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a^{2} c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.29, size = 117, normalized size = 1.89 \[ \frac {2\,\left (\frac {3\,B}{2}-A\,\cos \left (e+f\,x\right )+B\,\cos \left (e+f\,x\right )+2\,A\,\sin \left (e+f\,x\right )+B\,\sin \left (e+f\,x\right )-A\,\cos \left (2\,e+2\,f\,x\right )-\frac {B\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {A\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {B\,\sin \left (2\,e+2\,f\,x\right )}{2}\right )}{3\,a^2\,c\,f\,\left (2\,\cos \left (e+f\,x\right )+\sin \left (2\,e+2\,f\,x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.23, size = 578, normalized size = 9.32 \[ \begin {cases} - \frac {6 A \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 A \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 {2 A \tan {\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 {2 A}{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 B \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 {4 B \tan {\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 {2 B}{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 + B \sin {\relax (e )}\right )}{\left (a \sin {\relax (e )} + a\right )^{2} \left (- c \sin {\relax (e )} + c\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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