Optimal. Leaf size=53 \[ \frac {2 \tan (e+f x)}{3 a c^2 f}+\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2736, 2672, 3767, 8} \[ \frac {2 \tan (e+f x)}{3 a c^2 f}+\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2672
Rule 2736
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx &=\frac {\int \frac {\sec ^2(e+f x)}{c-c \sin (e+f x)} \, dx}{a c}\\ &=\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {2 \int \sec ^2(e+f x) \, dx}{3 a c^2}\\ &=\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}-\frac {2 \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a c^2 f}\\ &=\frac {\sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a c^2 f}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 87, normalized size = 1.64 \[ \frac {\sin (e+f x)+8 \sin (2 (e+f x))+\sin (3 (e+f x))+4 \cos (e+f x)-2 \cos (2 (e+f x))+4 \cos (3 (e+f x))-2}{24 a c^2 f (\sin (e+f x)-1)^2 (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 56, normalized size = 1.06 \[ -\frac {2 \, \cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1}{3 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 77, normalized size = 1.45 \[ -\frac {\frac {3}{a c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7}{a c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 73, normalized size = 1.38 \[ \frac {-\frac {2}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 142, normalized size = 2.68 \[ \frac {2 \, {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{3 \, {\left (a c^{2} - \frac {2 \, a c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a c^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.01, size = 74, normalized size = 1.40 \[ -\frac {2\,\left (3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}{3\,a\,c^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^3\,\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.02, size = 328, normalized size = 6.19 \[ \begin {cases} - \frac {6 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} + \frac {6 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\relax (e )} + a\right ) \left (- c \sin {\relax (e )} + c\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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