Optimal. Leaf size=80 \[ -\frac {\cot ^5(e+f x)}{5 a^3 c^2 f}+\frac {\csc ^5(e+f x)}{5 a^3 c^2 f}-\frac {2 \csc ^3(e+f x)}{3 a^3 c^2 f}+\frac {\csc (e+f x)}{a^3 c^2 f} \]
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Rubi [A] time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3958, 2606, 194, 2607, 30} \[ -\frac {\cot ^5(e+f x)}{5 a^3 c^2 f}+\frac {\csc ^5(e+f x)}{5 a^3 c^2 f}-\frac {2 \csc ^3(e+f x)}{3 a^3 c^2 f}+\frac {\csc (e+f x)}{a^3 c^2 f} \]
Antiderivative was successfully verified.
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Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 3958
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2} \, dx &=-\frac {\int \left (c \cot ^5(e+f x) \csc (e+f x)-c \cot ^4(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {\int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3 c^2}+\frac {\int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^2}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 c^2 f}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 c^2 f}\\ &=-\frac {\cot ^5(e+f x)}{5 a^3 c^2 f}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^2 f}\\ &=-\frac {\cot ^5(e+f x)}{5 a^3 c^2 f}+\frac {\csc (e+f x)}{a^3 c^2 f}-\frac {2 \csc ^3(e+f x)}{3 a^3 c^2 f}+\frac {\csc ^5(e+f x)}{5 a^3 c^2 f}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 147, normalized size = 1.84 \[ \frac {\csc (e) \sin ^2\left (\frac {1}{2} (e+f x)\right ) (-534 \sin (e+f x)-178 \sin (2 (e+f x))+178 \sin (3 (e+f x))+89 \sin (4 (e+f x))+40 \sin (2 e+f x)+168 \sin (e+2 f x)-120 \sin (3 e+2 f x)+72 \sin (2 e+3 f x)-120 \sin (4 e+3 f x)-24 \sin (3 e+4 f x)+200 \sin (e)+104 \sin (f x)) \csc ^5(e+f x)}{480 a^3 c^2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 109, normalized size = 1.36 \[ -\frac {3 \, \cos \left (f x + e\right )^{4} - 12 \, \cos \left (f x + e\right )^{3} - 12 \, \cos \left (f x + e\right )^{2} + 8 \, \cos \left (f x + e\right ) + 8}{15 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} + a^{3} c^{2} f \cos \left (f x + e\right )^{2} - a^{3} c^{2} f \cos \left (f x + e\right ) - a^{3} c^{2} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 108, normalized size = 1.35 \[ \frac {\frac {5 \, {\left (12 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}}{a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}} + \frac {3 \, a^{12} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 20 \, a^{12} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 90 \, a^{12} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{10}}}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.87, size = 76, normalized size = 0.95 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3}+6 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )-\frac {1}{3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {4}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{16 f \,a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 120, normalized size = 1.50 \[ \frac {\frac {\frac {90 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3} c^{2}} + \frac {5 \, {\left (\frac {12 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{a^{3} c^{2} \sin \left (f x + e\right )^{3}}}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 111, normalized size = 1.39 \[ \frac {48\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-192\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+168\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-32\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3}{240\,a^3\,c^2\,f\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} + \sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} - 2 \sec ^{2}{\left (e + f x \right )} + \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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