Optimal. Leaf size=210 \[ \frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot (e+f x)}{a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}+\frac {x}{a^3 c^5} \]
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Rubi [A] time = 0.24, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3904, 3886, 3473, 8, 2606, 194, 2607, 30} \[ \frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot (e+f x)}{a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}+\frac {x}{a^3 c^5} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 3473
Rule 3886
Rule 3904
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx &=-\frac {\int \cot ^{10}(e+f x) (a+a \sec (e+f x))^2 \, dx}{a^5 c^5}\\ &=-\frac {\int \left (a^2 \cot ^{10}(e+f x)+2 a^2 \cot ^9(e+f x) \csc (e+f x)+a^2 \cot ^8(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^5 c^5}\\ &=-\frac {\int \cot ^{10}(e+f x) \, dx}{a^3 c^5}-\frac {\int \cot ^8(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^5}-\frac {2 \int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^3 c^5}\\ &=\frac {\cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int \cot ^8(e+f x) \, dx}{a^3 c^5}-\frac {\operatorname {Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^3 c^5 f}+\frac {2 \operatorname {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f}\\ &=-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c^5}+\frac {2 \operatorname {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c^5}\\ &=-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c^5}\\ &=\frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int 1 \, dx}{a^3 c^5}\\ &=\frac {x}{a^3 c^5}+\frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}\\ \end {align*}
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Mathematica [B] time = 1.74, size = 441, normalized size = 2.10 \[ \frac {\csc \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}\right ) \tan (e+f x) \sec ^7(e+f x) (-1152405 \sin (e+f x)+512180 \sin (2 (e+f x))+486571 \sin (3 (e+f x))-409744 \sin (4 (e+f x))-25609 \sin (5 (e+f x))+102436 \sin (6 (e+f x))-25609 \sin (7 (e+f x))-825216 \sin (2 e+f x)+622976 \sin (e+2 f x)+142464 \sin (3 e+2 f x)+297088 \sin (2 e+3 f x)+430080 \sin (4 e+3 f x)-424192 \sin (3 e+4 f x)-188160 \sin (5 e+4 f x)+2048 \sin (4 e+5 f x)-40320 \sin (6 e+5 f x)+112768 \sin (5 e+6 f x)+40320 \sin (7 e+6 f x)-38272 \sin (6 e+7 f x)-453600 f x \cos (2 e+f x)-201600 f x \cos (e+2 f x)+201600 f x \cos (3 e+2 f x)-191520 f x \cos (2 e+3 f x)+191520 f x \cos (4 e+3 f x)+161280 f x \cos (3 e+4 f x)-161280 f x \cos (5 e+4 f x)+10080 f x \cos (4 e+5 f x)-10080 f x \cos (6 e+5 f x)-40320 f x \cos (5 e+6 f x)+40320 f x \cos (7 e+6 f x)+10080 f x \cos (6 e+7 f x)-10080 f x \cos (8 e+7 f x)+259584 \sin (e)-897024 \sin (f x)+453600 f x \cos (f x))}{2580480 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 271, normalized size = 1.29 \[ \frac {598 \, \cos \left (f x + e\right )^{7} - 566 \, \cos \left (f x + e\right )^{6} - 1212 \, \cos \left (f x + e\right )^{5} + 1310 \, \cos \left (f x + e\right )^{4} + 860 \, \cos \left (f x + e\right )^{3} - 1014 \, \cos \left (f x + e\right )^{2} + 315 \, {\left (f x \cos \left (f x + e\right )^{6} - 2 \, f x \cos \left (f x + e\right )^{5} - f x \cos \left (f x + e\right )^{4} + 4 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} - 2 \, f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) - 197 \, \cos \left (f x + e\right ) + 256}{315 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{6} - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} - a^{3} c^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{5} f \cos \left (f x + e\right )^{3} - a^{3} c^{5} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} c^{5} f \cos \left (f x + e\right ) + a^{3} c^{5} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 163, normalized size = 0.78 \[ \frac {\frac {40320 \, {\left (f x + e\right )}}{a^{3} c^{5}} + \frac {51345 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 9765 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 2331 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35}{a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} - \frac {63 \, {\left (a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 185 \, a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{25}}}{40320 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.98, size = 197, normalized size = 0.94 \[ -\frac {\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )}{640 f \,a^{3} c^{5}}+\frac {3 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{128 f \,a^{3} c^{5}}-\frac {37 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{128 f \,a^{3} c^{5}}+\frac {1}{1152 f \,a^{3} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}-\frac {9}{896 f \,a^{3} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {37}{640 f \,a^{3} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}-\frac {31}{128 f \,a^{3} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {163}{128 f \,a^{3} c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,a^{3} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 205, normalized size = 0.98 \[ -\frac {\frac {63 \, {\left (\frac {185 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {15 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{5}} - \frac {80640 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{5}} + \frac {{\left (\frac {405 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2331 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {9765 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {51345 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{3} c^{5} \sin \left (f x + e\right )^{9}}}{40320 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 233, normalized size = 1.11 \[ \frac {35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-63\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-11655\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+51345\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-9765\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+2331\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-405\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+40320\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (e+f\,x\right )}{40320\,a^3\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]
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 \[ - \frac {\int \frac {1}{\sec ^{8}{\left (e + f x \right )} - 2 \sec ^{7}{\left (e + f x \right )} - 2 \sec ^{6}{\left (e + f x \right )} + 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} - 1}\, dx}{a^{3} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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