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 = 0.236406, antiderivative size = 210, normalized size of antiderivative = 1., 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 3904
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
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*}
Mathematica [B] time = 1.72925, size = 441, normalized size = 2.1 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \tan (e+f x) \sec ^7(e+f x) (-1152405 \sin (e+f x)+512180 \sin (2 (e+f x))+486571 \sin (3 (e+f x))-409744 \sin (4 (e+f x))-25609 \sin (5 (e+f x))+102436 \sin (6 (e+f x))-25609 \sin (7 (e+f x))-825216 \sin (2 e+f x)+622976 \sin (e+2 f x)+142464 \sin (3 e+2 f x)+297088 \sin (2 e+3 f x)+430080 \sin (4 e+3 f x)-424192 \sin (3 e+4 f x)-188160 \sin (5 e+4 f x)+2048 \sin (4 e+5 f x)-40320 \sin (6 e+5 f x)+112768 \sin (5 e+6 f x)+40320 \sin (7 e+6 f x)-38272 \sin (6 e+7 f x)-453600 f x \cos (2 e+f x)-201600 f x \cos (e+2 f x)+201600 f x \cos (3 e+2 f x)-191520 f x \cos (2 e+3 f x)+191520 f x \cos (4 e+3 f x)+161280 f x \cos (3 e+4 f x)-161280 f x \cos (5 e+4 f x)+10080 f x \cos (4 e+5 f x)-10080 f x \cos (6 e+5 f x)-40320 f x \cos (5 e+6 f x)+40320 f x \cos (7 e+6 f x)+10080 f x \cos (6 e+7 f x)-10080 f x \cos (8 e+7 f x)+259584 \sin (e)-897024 \sin (f x)+453600 f x \cos (f x))}{2580480 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 197, normalized size = 0.9 \begin{align*} -{\frac{1}{640\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{3}{128\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{37}{128\,f{a}^{3}{c}^{5}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}{c}^{5}}}+{\frac{1}{1152\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}}-{\frac{9}{896\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{37}{640\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{31}{128\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+{\frac{163}{128\,f{a}^{3}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54884, size = 277, normalized size = 1.32 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16328, size = 687, normalized size = 3.27 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43953, size = 220, normalized size = 1.05 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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