Optimal. Leaf size=78 \[ \frac {2 \cot ^5(e+f x)}{5 a c^3 f}+\frac {2 \csc ^5(e+f x)}{5 a c^3 f}-\frac {\csc ^3(e+f x)}{a c^3 f}+\frac {\csc (e+f x)}{a c^3 f} \]
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Rubi [A] time = 0.18, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3958, 2606, 194, 2607, 30, 14} \[ \frac {2 \cot ^5(e+f x)}{5 a c^3 f}+\frac {2 \csc ^5(e+f x)}{5 a c^3 f}-\frac {\csc ^3(e+f x)}{a c^3 f}+\frac {\csc (e+f x)}{a c^3 f} \]
Antiderivative was successfully verified.
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Rule 14
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)) (c-c \sec (e+f x))^3} \, dx &=-\frac {\int \left (a^2 \cot ^5(e+f x) \csc (e+f x)+2 a^2 \cot ^4(e+f x) \csc ^2(e+f x)+a^2 \cot ^3(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {\int \cot ^5(e+f x) \csc (e+f x) \, dx}{a c^3}-\frac {\int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a c^3}-\frac {2 \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a c^3}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a c^3 f}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a c^3 f}-\frac {2 \operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a c^3 f}\\ &=\frac {2 \cot ^5(e+f x)}{5 a c^3 f}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a c^3 f}+\frac {\operatorname {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a c^3 f}\\ &=\frac {2 \cot ^5(e+f x)}{5 a c^3 f}+\frac {\csc (e+f x)}{a c^3 f}-\frac {\csc ^3(e+f x)}{a c^3 f}+\frac {2 \csc ^5(e+f x)}{5 a c^3 f}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 107, normalized size = 1.37 \[ -\frac {\csc (e) (65 \sin (e+f x)-52 \sin (2 (e+f x))+13 \sin (3 (e+f x))+40 \sin (2 e+f x)-12 \sin (e+2 f x)-20 \sin (3 e+2 f x)+8 \sin (2 e+3 f x)-40 \sin (e)) \csc ^4\left (\frac {1}{2} (e+f x)\right ) \csc (e+f x)}{320 a c^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 74, normalized size = 0.95 \[ \frac {2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - 4 \, \cos \left (f x + e\right ) + 2}{5 \, {\left (a c^{3} f \cos \left (f x + e\right )^{2} - 2 \, a c^{3} f \cos \left (f x + e\right ) + a c^{3} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 73, normalized size = 0.94 \[ \frac {\frac {5 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a c^{3}} + \frac {15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 5 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1}{a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 61, normalized size = 0.78 \[ \frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-\frac {1}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {1}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}+\frac {3}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{8 f a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 97, normalized size = 1.24 \[ -\frac {\frac {{\left (\frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{a c^{3} \sin \left (f x + e\right )^{5}} - \frac {5 \, \sin \left (f x + e\right )}{a c^{3} {\left (\cos \left (f x + e\right ) + 1\right )}}}{40 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 63, normalized size = 0.81 \[ \frac {5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1}{40\,a\,c^3\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]
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 \[ - \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} - 1}\, dx}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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