Optimal. Leaf size=120 \[ \frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{\csc ^7(e+f x)}{a^3 c^5 f}+\frac{9 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{5 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{\csc (e+f x)}{a^3 c^5 f} \]
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Rubi [A] time = 0.204163, antiderivative size = 120, normalized size of antiderivative = 1., 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} \[ \frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac{\csc ^7(e+f x)}{a^3 c^5 f}+\frac{9 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{5 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{\csc (e+f x)}{a^3 c^5 f} \]
Antiderivative was successfully verified.
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Rule 3958
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 270
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx &=-\frac{\int \left (a^2 \cot ^9(e+f x) \csc (e+f x)+2 a^2 \cot ^8(e+f x) \csc ^2(e+f x)+a^2 \cot ^7(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^5 c^5}\\ &=-\frac{\int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^3 c^5}-\frac{\int \cot ^7(e+f x) \csc ^3(e+f x) \, dx}{a^3 c^5}-\frac{2 \int \cot ^8(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^5}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f}-\frac{2 \operatorname{Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^3 c^5 f}\\ &=\frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{\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{\operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f}\\ &=\frac{2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac{\csc (e+f x)}{a^3 c^5 f}-\frac{5 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac{9 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac{\csc ^7(e+f x)}{a^3 c^5 f}+\frac{2 \csc ^9(e+f x)}{9 a^3 c^5 f}\\ \end{align*}
Mathematica [B] time = 1.56247, size = 257, normalized size = 2.14 \[ -\frac{\csc (e) (76455 \sin (e+f x)-33980 \sin (2 (e+f x))-32281 \sin (3 (e+f x))+27184 \sin (4 (e+f x))+1699 \sin (5 (e+f x))-6796 \sin (6 (e+f x))+1699 \sin (7 (e+f x))+22656 \sin (2 e+f x)-17216 \sin (e+2 f x)+4416 \sin (3 e+2 f x)+3200 \sin (2 e+3 f x)-15360 \sin (4 e+3 f x)+12160 \sin (3 e+4 f x)-1920 \sin (5 e+4 f x)-5120 \sin (4 e+5 f x)+5760 \sin (6 e+5 f x)+320 \sin (5 e+6 f x)-2880 \sin (7 e+6 f x)+640 \sin (6 e+7 f x)-33024 \sin (e)+6144 \sin (f x)) \tan (e+f x) \sec ^7(e+f x)}{184320 a^3 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 115, normalized size = 1. \begin{align*}{\frac{1}{128\,f{a}^{3}{c}^{5}} \left ({\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{7}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+21\,\tan \left ( 1/2\,fx+e/2 \right ) -{\frac{35}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+35\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{-1}+{\frac{21}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}- \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}+{\frac{1}{9} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03045, size = 244, normalized size = 2.03 \begin{align*} \frac{\frac{3 \,{\left (\frac{315 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{35 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{5}} - \frac{{\left (\frac{45 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{189 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{525 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{1575 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 5\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{3} c^{5} \sin \left (f x + e\right )^{9}}}{5760 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.476116, size = 462, normalized size = 3.85 \begin{align*} \frac{10 \, \cos \left (f x + e\right )^{7} + 25 \, \cos \left (f x + e\right )^{6} - 60 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{4} + 80 \, \cos \left (f x + e\right )^{3} - 24 \, \cos \left (f x + e\right )^{2} - 32 \, \cos \left (f x + e\right ) + 16}{45 \,{\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 = 1.40693, size = 203, normalized size = 1.69 \begin{align*} \frac{\frac{1575 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 525 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 189 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 45 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 5}{a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}} + \frac{3 \,{\left (3 \, a^{12} c^{20} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 35 \, a^{12} c^{20} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 315 \, a^{12} c^{20} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15} c^{25}}}{5760 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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