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 = 0.22, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 2648
Rule 2735
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))^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*}
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Mathematica [B] time = 0.63, 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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fricas [B] time = 0.44, size = 162, normalized size = 2.25 \[ -\frac {6 \, B a f x - {\left (3 \, B a f x + {\left (A + 7 \, B\right )} a\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (A + B\right )} a + {\left (3 \, B a f x + {\left (A - 5 \, B\right )} a\right )} \cos \left (f x + e\right ) - {\left (6 \, B a f x - 2 \, {\left (A + B\right )} a + {\left (3 \, B a f x - {\left (A + 7 \, B\right )} a\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f + {\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 92, normalized size = 1.28 \[ \frac {\frac {3 \, {\left (f x + e\right )} B a}{c^{2}} - \frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + A a - 5 \, B a\right )}}{c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 160, normalized size = 2.22 \[ -\frac {2 a A}{c^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {2 a B}{c^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {8 a A}{3 c^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {8 a B}{3 c^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4 a A}{c^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {4 a B}{c^{2} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}+\frac {2 a B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 456, normalized size = 6.33 \[ \frac {2 \, {\left (B a {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 4}{c^{2} - \frac {3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{2}}\right )} - \frac {A a {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 2\right )}}{c^{2} - \frac {3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {A a {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}}{c^{2} - \frac {3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {B a {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}}{c^{2} - \frac {3 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.51, size = 132, normalized size = 1.83 \[ \frac {B\,a\,x}{c^2}-\frac {\left (\frac {a\,\left (6\,A-6\,B+9\,B\,\left (e+f\,x\right )\right )}{3}-3\,B\,a\,\left (e+f\,x\right )\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\left (\frac {a\,\left (24\,B-9\,B\,\left (e+f\,x\right )\right )}{3}+3\,B\,a\,\left (e+f\,x\right )\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {a\,\left (2\,A-10\,B+3\,B\,\left (e+f\,x\right )\right )}{3}-B\,a\,\left (e+f\,x\right )}{c^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.11, size = 700, normalized size = 9.72 \[ \begin {cases} - \frac {6 A a \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} - \frac {2 A a}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} + \frac {3 B a f x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} - \frac {9 B a f x \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} + \frac {9 B a f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} - \frac {3 B a f x}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} + \frac {6 B a \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} - \frac {24 B a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} + \frac {10 B a}{3 c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 9 c^{2} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 c^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right )}{\left (- c \sin {\relax (e )} + c\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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