Optimal. Leaf size=80 \[ \frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f} \]
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Rubi [A] time = 0.15, 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} \[ \frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f} \]
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))^2 (c-c \sec (e+f x))^3} \, dx &=-\frac {\int \left (a \cot ^5(e+f x) \csc (e+f x)+a \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^2 c^3}-\frac {\int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^2 c^3}\\ &=-\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^2 c^3 f}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^2 c^3 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^3 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^2 c^3 f}+\frac {\csc (e+f x)}{a^2 c^3 f}-\frac {2 \csc ^3(e+f x)}{3 a^2 c^3 f}+\frac {\csc ^5(e+f x)}{5 a^2 c^3 f}\\ \end {align*}
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Mathematica [A] time = 0.96, size = 147, normalized size = 1.84 \[ -\frac {\csc (e) (534 \sin (e+f x)-178 \sin (2 (e+f x))-178 \sin (3 (e+f x))+89 \sin (4 (e+f x))+40 \sin (2 e+f x)-168 \sin (e+2 f x)+120 \sin (3 e+2 f x)+72 \sin (2 e+3 f x)-120 \sin (4 e+3 f x)+24 \sin (3 e+4 f x)-200 \sin (e)+104 \sin (f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \csc ^3(e+f x)}{1920 a^2 c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 109, normalized size = 1.36 \[ \frac {3 \, \cos \left (f x + e\right )^{4} + 12 \, \cos \left (f x + e\right )^{3} - 12 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 8}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 101, normalized size = 1.26 \[ \frac {\frac {90 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3}{a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} - \frac {5 \, {\left (a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{9}}}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 76, normalized size = 0.95 \[ \frac {-\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3}+4 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )-\frac {4}{3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {1}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}+\frac {6}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{16 f \,a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 121, normalized size = 1.51 \[ \frac {\frac {5 \, {\left (\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{3}} - \frac {{\left (\frac {20 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {90 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{a^{2} c^{3} \sin \left (f x + e\right )^{5}}}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.01, size = 76, normalized size = 0.95 \[ \frac {-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+60\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3}{240\,a^2\,c^3\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]
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 \[ - \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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