Optimal. Leaf size=59 \[ \frac{\cot ^3(e+f x)}{3 a^2 c f}-\frac{\csc ^3(e+f x)}{3 a^2 c f}+\frac{\csc (e+f x)}{a^2 c f} \]
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Rubi [A] time = 0.136156, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3958, 2606, 2607, 30} \[ \frac{\cot ^3(e+f x)}{3 a^2 c f}-\frac{\csc ^3(e+f x)}{3 a^2 c f}+\frac{\csc (e+f x)}{a^2 c f} \]
Antiderivative was successfully verified.
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Rule 3958
Rule 2606
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx &=\frac{\int \left (c \cot ^3(e+f x) \csc (e+f x)-c \cot ^2(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^2 c^2}\\ &=\frac{\int \cot ^3(e+f x) \csc (e+f x) \, dx}{a^2 c}-\frac{\int \cot ^2(e+f x) \csc ^2(e+f x) \, dx}{a^2 c}\\ &=-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (e+f x)\right )}{a^2 c f}-\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c f}\\ &=\frac{\cot ^3(e+f x)}{3 a^2 c f}+\frac{\csc (e+f x)}{a^2 c f}-\frac{\csc ^3(e+f x)}{3 a^2 c f}\\ \end{align*}
Mathematica [A] time = 0.570834, size = 83, normalized size = 1.41 \[ -\frac{\csc (e) \sin ^2\left (\frac{1}{2} (e+f x)\right ) (10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)-2 \sin (e+2 f x)-6 \sin (e)+2 \sin (f x)) \csc ^3(e+f x)}{6 a^2 c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 48, normalized size = 0.8 \begin{align*}{\frac{1}{4\,f{a}^{2}c} \left ( -{\frac{1}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+2\,\tan \left ( 1/2\,fx+e/2 \right ) + \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988689, size = 103, normalized size = 1.75 \begin{align*} \frac{\frac{\frac{6 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2} c} + \frac{3 \,{\left (\cos \left (f x + e\right ) + 1\right )}}{a^{2} c \sin \left (f x + e\right )}}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.438719, size = 124, normalized size = 2.1 \begin{align*} -\frac{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 2}{3 \,{\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c 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 ^{3}{\left (e + f x \right )} + \sec ^{2}{\left (e + f x \right )} - \sec{\left (e + f x \right )} - 1}\, dx}{a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22861, size = 97, normalized size = 1.64 \begin{align*} \frac{\frac{3}{a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )} - \frac{a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 6 \, a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{6} c^{3}}}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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