Optimal. Leaf size=99 \[ -\frac {\cot ^7(e+f x)}{7 a^3 c^4 f}-\frac {\csc ^7(e+f x)}{7 a^3 c^4 f}+\frac {3 \csc ^5(e+f x)}{5 a^3 c^4 f}-\frac {\csc ^3(e+f x)}{a^3 c^4 f}+\frac {\csc (e+f x)}{a^3 c^4 f} \]
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Rubi [A] time = 0.15, antiderivative size = 99, 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} \[ -\frac {\cot ^7(e+f x)}{7 a^3 c^4 f}-\frac {\csc ^7(e+f x)}{7 a^3 c^4 f}+\frac {3 \csc ^5(e+f x)}{5 a^3 c^4 f}-\frac {\csc ^3(e+f x)}{a^3 c^4 f}+\frac {\csc (e+f x)}{a^3 c^4 f} \]
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))^4} \, dx &=\frac {\int \left (a \cot ^7(e+f x) \csc (e+f x)+a \cot ^6(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^4 c^4}\\ &=\frac {\int \cot ^7(e+f x) \csc (e+f x) \, dx}{a^3 c^4}+\frac {\int \cot ^6(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^4}\\ &=\frac {\operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (e+f x)\right )}{a^3 c^4 f}-\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^3 c^4 f}\\ &=-\frac {\cot ^7(e+f x)}{7 a^3 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^3 c^4 f}\\ &=-\frac {\cot ^7(e+f x)}{7 a^3 c^4 f}+\frac {\csc (e+f x)}{a^3 c^4 f}-\frac {\csc ^3(e+f x)}{a^3 c^4 f}+\frac {3 \csc ^5(e+f x)}{5 a^3 c^4 f}-\frac {\csc ^7(e+f x)}{7 a^3 c^4 f}\\ \end {align*}
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Mathematica [B] time = 1.41, size = 211, normalized size = 2.13 \[ \frac {\csc (e) (-7620 \sin (e+f x)+1905 \sin (2 (e+f x))+3810 \sin (3 (e+f x))-1524 \sin (4 (e+f x))-762 \sin (5 (e+f x))+381 \sin (6 (e+f x))-2016 \sin (2 e+f x)+2080 \sin (e+2 f x)-1680 \sin (3 e+2 f x)+240 \sin (2 e+3 f x)+560 \sin (4 e+3 f x)-880 \sin (3 e+4 f x)+560 \sin (5 e+4 f x)+400 \sin (4 e+5 f x)-560 \sin (6 e+5 f x)+80 \sin (5 e+6 f x)+2912 \sin (e)+416 \sin (f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \csc ^5(e+f x)}{35840 a^3 c^4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 163, normalized size = 1.65 \[ \frac {5 \, \cos \left (f x + e\right )^{6} + 30 \, \cos \left (f x + e\right )^{5} - 30 \, \cos \left (f x + e\right )^{4} - 40 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} + 16 \, \cos \left (f x + e\right ) - 16}{35 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{5} - a^{3} c^{4} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{4} f \cos \left (f x + e\right )^{3} + 2 \, a^{3} c^{4} f \cos \left (f x + e\right )^{2} + a^{3} c^{4} f \cos \left (f x + e\right ) - a^{3} c^{4} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 135, normalized size = 1.36 \[ \frac {\frac {700 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 175 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 42 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5}{a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} + \frac {7 \, {\left (a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 75 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{20}}}{2240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.93, size = 102, normalized size = 1.03 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5}-2 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )+15 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )-\frac {1}{7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}-\frac {5}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {6}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}+\frac {20}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{64 f \,a^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 159, normalized size = 1.61 \[ \frac {\frac {7 \, {\left (\frac {75 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{4}} + \frac {{\left (\frac {42 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {175 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {700 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{a^{3} c^{4} \sin \left (f x + e\right )^{7}}}{2240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 129, normalized size = 1.30 \[ \frac {\left (2\,{\sin \left (\frac {e}{4}+\frac {f\,x}{4}\right )}^2-1\right )\,\left (\frac {235\,{\sin \left (e+f\,x\right )}^2}{16}-\frac {45\,{\sin \left (2\,e+2\,f\,x\right )}^2}{8}+\frac {19\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{2}+\frac {5\,{\sin \left (3\,e+3\,f\,x\right )}^2}{16}-\frac {5\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2}{4}+\frac {15\,{\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}^2}{4}-5\right )}{2240\,a^3\,c^4\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,{\left ({\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^3} \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - \sec ^{6}{\left (e + f x \right )} - 3 \sec ^{5}{\left (e + f x \right )} + 3 \sec ^{4}{\left (e + f x \right )} + 3 \sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} - \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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