Optimal. Leaf size=76 \[ \frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2736, 2672, 3767} \[ \frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2736
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx &=\frac {\int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)} \, dx}{a^2 c^2}\\ &=\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {4 \int \sec ^4(e+f x) \, dx}{5 a^2 c^3}\\ &=\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {4 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{5 a^2 c^3 f}\\ &=\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 131, normalized size = 1.72 \[ -\frac {18 \sin (e+f x)+512 \sin (2 (e+f x))+27 \sin (3 (e+f x))+128 \sin (4 (e+f x))+9 \sin (5 (e+f x))+128 \cos (e+f x)-72 \cos (2 (e+f x))+192 \cos (3 (e+f x))-18 \cos (4 (e+f x))+64 \cos (5 (e+f x))-54}{1920 a^2 c^3 f (\sin (e+f x)-1)^3 (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 86, normalized size = 1.13 \[ -\frac {8 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sin \left (f x + e\right ) - 1}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a^{2} c^{3} f \cos \left (f x + e\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 133, normalized size = 1.75 \[ -\frac {\frac {5 \, {\left (15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13\right )}}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {165 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 480 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 650 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 400 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 113}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 133, normalized size = 1.75 \[ \frac {-\frac {2}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {5}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {5}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,c^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 335, normalized size = 4.41 \[ \frac {2 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {13 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {25 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 3\right )}}{15 \, {\left (a^{2} c^{3} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a^{2} c^{3} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.83, size = 128, normalized size = 1.68 \[ -\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+25\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+13\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+9\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+3\right )}{15\,a^2\,c^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^5\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.88, size = 1418, normalized size = 18.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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