Optimal. Leaf size=78 \[ \frac{2 \cot ^5(e+f x)}{5 a c^3 f}+\frac{2 \csc ^5(e+f x)}{5 a c^3 f}-\frac{\csc ^3(e+f x)}{a c^3 f}+\frac{\csc (e+f x)}{a c^3 f} \]
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Rubi [A] time = 0.179323, antiderivative size = 78, 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, 14} \[ \frac{2 \cot ^5(e+f x)}{5 a c^3 f}+\frac{2 \csc ^5(e+f x)}{5 a c^3 f}-\frac{\csc ^3(e+f x)}{a c^3 f}+\frac{\csc (e+f x)}{a c^3 f} \]
Antiderivative was successfully verified.
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Rule 3958
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 14
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^3} \, dx &=-\frac{\int \left (a^2 \cot ^5(e+f x) \csc (e+f x)+2 a^2 \cot ^4(e+f x) \csc ^2(e+f x)+a^2 \cot ^3(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac{\int \cot ^5(e+f x) \csc (e+f x) \, dx}{a c^3}-\frac{\int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a c^3}-\frac{2 \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a c^3}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a c^3 f}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a c^3 f}-\frac{2 \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a c^3 f}\\ &=\frac{2 \cot ^5(e+f x)}{5 a c^3 f}+\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a c^3 f}+\frac{\operatorname{Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a c^3 f}\\ &=\frac{2 \cot ^5(e+f x)}{5 a c^3 f}+\frac{\csc (e+f x)}{a c^3 f}-\frac{\csc ^3(e+f x)}{a c^3 f}+\frac{2 \csc ^5(e+f x)}{5 a c^3 f}\\ \end{align*}
Mathematica [A] time = 0.866247, size = 107, normalized size = 1.37 \[ -\frac{\csc (e) (65 \sin (e+f x)-52 \sin (2 (e+f x))+13 \sin (3 (e+f x))+40 \sin (2 e+f x)-12 \sin (e+2 f x)-20 \sin (3 e+2 f x)+8 \sin (2 e+3 f x)-40 \sin (e)) \csc ^4\left (\frac{1}{2} (e+f x)\right ) \csc (e+f x)}{320 a c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 61, normalized size = 0.8 \begin{align*}{\frac{1}{8\,fa{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) - \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}+3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{-1}+{\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983902, size = 131, normalized size = 1.68 \begin{align*} -\frac{\frac{{\left (\frac{5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{a c^{3} \sin \left (f x + e\right )^{5}} - \frac{5 \, \sin \left (f x + e\right )}{a c^{3}{\left (\cos \left (f x + e\right ) + 1\right )}}}{40 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.445246, size = 185, normalized size = 2.37 \begin{align*} \frac{2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - 4 \, \cos \left (f x + e\right ) + 2}{5 \,{\left (a c^{3} f \cos \left (f x + e\right )^{2} - 2 \, a c^{3} f \cos \left (f x + e\right ) + a c^{3} 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] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sec{\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} - 1}\, dx}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3243, size = 99, normalized size = 1.27 \begin{align*} \frac{\frac{5 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a c^{3}} + \frac{15 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 5 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1}{a c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}}}{40 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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