Optimal. Leaf size=98 \[ -\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac {\csc ^5(e+f x)}{a^2 c^4 f}-\frac {4 \csc ^3(e+f x)}{3 a^2 c^4 f}+\frac {\csc (e+f x)}{a^2 c^4 f} \]
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Rubi [A] time = 0.19, antiderivative size = 98, 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, 270} \[ -\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac {\csc ^5(e+f x)}{a^2 c^4 f}-\frac {4 \csc ^3(e+f x)}{3 a^2 c^4 f}+\frac {\csc (e+f x)}{a^2 c^4 f} \]
Antiderivative was successfully verified.
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Rule 30
Rule 194
Rule 270
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))^4} \, dx &=\frac {\int \left (a^2 \cot ^7(e+f x) \csc (e+f x)+2 a^2 \cot ^6(e+f x) \csc ^2(e+f x)+a^2 \cot ^5(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^4 c^4}\\ &=\frac {\int \cot ^7(e+f x) \csc (e+f x) \, dx}{a^2 c^4}+\frac {\int \cot ^5(e+f x) \csc ^3(e+f x) \, dx}{a^2 c^4}+\frac {2 \int \cot ^6(e+f x) \csc ^2(e+f x) \, dx}{a^2 c^4}\\ &=-\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^2 c^4 f}-\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^2 c^4 f}+\frac {2 \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (e+f x)\right )}{a^2 c^4 f}\\ &=-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}-\frac {\operatorname {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^4 f}-\frac {\operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^4 f}\\ &=-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {\csc (e+f x)}{a^2 c^4 f}-\frac {4 \csc ^3(e+f x)}{3 a^2 c^4 f}+\frac {\csc ^5(e+f x)}{a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 179, normalized size = 1.83 \[ \frac {\csc (e) (-182 \sin (e+f x)+104 \sin (2 (e+f x))+39 \sin (3 (e+f x))-52 \sin (4 (e+f x))+13 \sin (5 (e+f x))-56 \sin (2 e+f x)+76 \sin (e+2 f x)-28 \sin (3 e+2 f x)-24 \sin (2 e+3 f x)+42 \sin (4 e+3 f x)-3 \sin (3 e+4 f x)-21 \sin (5 e+4 f x)+6 \sin (4 e+5 f x)+42 \sin (e)-28 \sin (f x)) \csc ^4\left (\frac {1}{2} (e+f x)\right ) \csc ^3(e+f x)}{1344 a^2 c^4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 120, normalized size = 1.22 \[ \frac {6 \, \cos \left (f x + e\right )^{5} + 9 \, \cos \left (f x + e\right )^{4} - 24 \, \cos \left (f x + e\right )^{3} + 4 \, \cos \left (f x + e\right )^{2} + 16 \, \cos \left (f x + e\right ) - 8}{21 \, {\left (a^{2} c^{4} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} c^{4} f \cos \left (f x + e\right ) - a^{2} c^{4} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 115, normalized size = 1.17 \[ \frac {\frac {210 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 70 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 21 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3}{a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} - \frac {7 \, {\left (a^{4} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{4} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{12}}}{672 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 87, normalized size = 0.89 \[ \frac {-\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3}+5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )-\frac {1}{7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}-\frac {10}{3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {1}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}+\frac {10}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{32 f \,a^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 140, normalized size = 1.43 \[ \frac {\frac {7 \, {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{4}} + \frac {{\left (\frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {70 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {210 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{a^{2} c^{4} \sin \left (f x + e\right )^{7}}}{672 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 89, normalized size = 0.91 \[ \frac {-7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+105\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+210\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-70\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-3}{672\,a^2\,c^4\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7} \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{6}{\left (e + f x \right )} - 2 \sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} + 4 \sec ^{3}{\left (e + f x \right )} - \sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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