Optimal. Leaf size=118 \[ -\frac {a^2 c^2 (A-B) \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac {3 c^2 (2 A-3 B) \cos (e+f x)}{2 a f}-\frac {c^2 (2 A-3 B) \cos ^3(e+f x)}{2 f (a \sin (e+f x)+a)}-\frac {3 c^2 x (2 A-3 B)}{2 a} \]
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Rubi [A] time = 0.28, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2967, 2859, 2679, 2682, 8} \[ -\frac {a^2 c^2 (A-B) \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^3}-\frac {3 c^2 (2 A-3 B) \cos (e+f x)}{2 a f}-\frac {c^2 (2 A-3 B) \cos ^3(e+f x)}{2 f (a \sin (e+f x)+a)}-\frac {3 c^2 x (2 A-3 B)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2679
Rule 2682
Rule 2859
Rule 2967
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac {a^2 (A-B) c^2 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^3}-\left (a (2 A-3 B) c^2\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx\\ &=-\frac {a^2 (A-B) c^2 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^3}-\frac {(2 A-3 B) c^2 \cos ^3(e+f x)}{2 f (a+a \sin (e+f x))}-\frac {1}{2} \left (3 (2 A-3 B) c^2\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx\\ &=-\frac {3 (2 A-3 B) c^2 \cos (e+f x)}{2 a f}-\frac {a^2 (A-B) c^2 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^3}-\frac {(2 A-3 B) c^2 \cos ^3(e+f x)}{2 f (a+a \sin (e+f x))}-\frac {\left (3 (2 A-3 B) c^2\right ) \int 1 \, dx}{2 a}\\ &=-\frac {3 (2 A-3 B) c^2 x}{2 a}-\frac {3 (2 A-3 B) c^2 \cos (e+f x)}{2 a f}-\frac {a^2 (A-B) c^2 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^3}-\frac {(2 A-3 B) c^2 \cos ^3(e+f x)}{2 f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 1.33, size = 188, normalized size = 1.59 \[ -\frac {c^2 (\sin (e+f x)-1)^2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right ) (6 (2 A-3 B) (e+f x)+4 (A-3 B) \cos (e+f x)+B \sin (2 (e+f x)))+\sin \left (\frac {1}{2} (e+f x)\right ) (4 (A-3 B) \cos (e+f x)+4 A (3 e+3 f x-8)-2 B (9 e+9 f x-16)+B \sin (2 (e+f x)))\right )}{4 a f (\sin (e+f x)+1) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 179, normalized size = 1.52 \[ \frac {B c^{2} \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, A - 3 \, B\right )} c^{2} f x - 2 \, {\left (A - 3 \, B\right )} c^{2} \cos \left (f x + e\right )^{2} - 8 \, {\left (A - B\right )} c^{2} - {\left (3 \, {\left (2 \, A - 3 \, B\right )} c^{2} f x + {\left (10 \, A - 13 \, B\right )} c^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, {\left (2 \, A - 3 \, B\right )} c^{2} f x + B c^{2} \cos \left (f x + e\right )^{2} + {\left (2 \, A - 5 \, B\right )} c^{2} \cos \left (f x + e\right ) - 8 \, {\left (A - B\right )} c^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 164, normalized size = 1.39 \[ -\frac {\frac {3 \, {\left (2 \, A c^{2} - 3 \, B c^{2}\right )} {\left (f x + e\right )}}{a} + \frac {16 \, {\left (A c^{2} - B c^{2}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} - \frac {2 \, {\left (B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, A c^{2} + 6 \, B c^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 299, normalized size = 2.53 \[ \frac {c^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) B}{f a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {2 c^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) A}{f a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {6 c^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) B}{f a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {c^{2} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {2 c^{2} A}{f a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {6 c^{2} B}{f a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {9 c^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) B}{f a}-\frac {6 c^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) A}{f a}-\frac {8 c^{2} A}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {8 c^{2} B}{f a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 608, normalized size = 5.15 \[ \frac {B c^{2} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \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}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 2 \, A c^{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}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \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 )}{a}\right )} + 4 \, B c^{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}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \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 )}{a}\right )} - 4 \, A c^{2} {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + 2 \, B c^{2} {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {2 \, A c^{2}}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.58, size = 241, normalized size = 2.04 \[ -\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A\,c^2-5\,B\,c^2\right )+10\,A\,c^2-14\,B\,c^2+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,A\,c^2-7\,B\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (8\,A\,c^2-9\,B\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (18\,A\,c^2-21\,B\,c^2\right )}{f\,\left (a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\right )}-\frac {3\,c^2\,\mathrm {atan}\left (\frac {3\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A-3\,B\right )}{6\,A\,c^2-9\,B\,c^2}\right )\,\left (2\,A-3\,B\right )}{a\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.10, size = 2365, normalized size = 20.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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