Optimal. Leaf size=84 \[ \frac {A \tan ^5(e+f x)}{5 a^3 c^3 f}+\frac {2 A \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {A \tan (e+f x)}{a^3 c^3 f}+\frac {B \sec ^5(e+f x)}{5 a^3 c^3 f} \]
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Rubi [A] time = 0.15, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2967, 2669, 3767} \[ \frac {A \tan ^5(e+f x)}{5 a^3 c^3 f}+\frac {2 A \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {A \tan (e+f x)}{a^3 c^3 f}+\frac {B \sec ^5(e+f x)}{5 a^3 c^3 f} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2967
Rule 3767
Rubi steps
\begin {align*} \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx &=\frac {\int \sec ^6(e+f x) (A+B \sin (e+f x)) \, dx}{a^3 c^3}\\ &=\frac {B \sec ^5(e+f x)}{5 a^3 c^3 f}+\frac {A \int \sec ^6(e+f x) \, dx}{a^3 c^3}\\ &=\frac {B \sec ^5(e+f x)}{5 a^3 c^3 f}-\frac {A \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{a^3 c^3 f}\\ &=\frac {B \sec ^5(e+f x)}{5 a^3 c^3 f}+\frac {A \tan (e+f x)}{a^3 c^3 f}+\frac {2 A \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {A \tan ^5(e+f x)}{5 a^3 c^3 f}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 65, normalized size = 0.77 \[ \frac {A \left (\frac {1}{5} \tan ^5(e+f x)+\frac {2}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{a^3 c^3 f}+\frac {B \sec ^5(e+f x)}{5 a^3 c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 56, normalized size = 0.67 \[ \frac {{\left (8 \, A \cos \left (f x + e\right )^{4} + 4 \, A \cos \left (f x + e\right )^{2} + 3 \, A\right )} \sin \left (f x + e\right ) + 3 \, B}{15 \, a^{3} c^{3} f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 134, normalized size = 1.60 \[ -\frac {2 \, {\left (15 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 20 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 58 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B\right )}}{15 \, {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{5} a^{3} c^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 227, normalized size = 2.70 \[ \frac {-\frac {A +B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {A}{2}+\frac {B}{2}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {\frac {7 A}{8}+\frac {5 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {A}{2}+\frac {3 B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {11 A}{8}+\frac {9 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {-A +B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {-\frac {7 A}{8}+\frac {5 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{2}-\frac {B}{2}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {A}{2}-\frac {3 B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (\frac {11 A}{8}-\frac {9 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 60, normalized size = 0.71 \[ \frac {\frac {{\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} A}{a^{3} c^{3}} + \frac {3 \, B}{a^{3} c^{3} \cos \left (f x + e\right )^{5}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.42, size = 126, normalized size = 1.50 \[ -\frac {2\,\left (15\,A\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+15\,B\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,A\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+58\,A\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+30\,B\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-20\,A\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+15\,A\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+3\,B\right )}{15\,a^3\,c^3\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.93, size = 1098, normalized size = 13.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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