Optimal. Leaf size=83 \[ \frac{\tan (e+f x)}{f (c-d) (a \sec (e+f x)+a)}-\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{a f (c-d)^{3/2} \sqrt{c+d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155081, antiderivative size = 134, normalized size of antiderivative = 1.61, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3987, 96, 93, 205} \[ \frac{2 d \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}{\sqrt{c-d} \sqrt{a-a \sec (e+f x)}}\right )}{f (c-d)^{3/2} \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\tan (e+f x)}{f (c-d) (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3987
Rule 96
Rule 93
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c+d \sec (e+f x))} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\tan (e+f x)}{(c-d) f (a+a \sec (e+f x))}+\frac{(a d \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} \sqrt{a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{(c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\tan (e+f x)}{(c-d) f (a+a \sec (e+f x))}+\frac{(2 a d \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{a-a \sec (e+f x)}}\right )}{(c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\tan (e+f x)}{(c-d) f (a+a \sec (e+f x))}+\frac{2 d \tan ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+a \sec (e+f x)}}{\sqrt{c-d} \sqrt{a-a \sec (e+f x)}}\right ) \tan (e+f x)}{(c-d)^{3/2} \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.663608, size = 160, normalized size = 1.93 \[ \frac{2 \cos \left (\frac{1}{2} (e+f x)\right ) \left (\sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right )+\frac{2 d (\sin (e)+i \cos (e)) \cos \left (\frac{1}{2} (e+f x)\right ) \tan ^{-1}\left (\frac{(\sin (e)+i \cos (e)) \left (\tan \left (\frac{f x}{2}\right ) (c \cos (e)-d)+c \sin (e)\right )}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right )}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right )}{a f (c-d) (\cos (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.065, size = 74, normalized size = 0.9 \begin{align*}{\frac{1}{fa} \left ({\frac{1}{c-d}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }-2\,{\frac{d}{ \left ( c-d \right ) \sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}{\it Artanh} \left ({\frac{\tan \left ( 1/2\,fx+e/2 \right ) \left ( c-d \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.505809, size = 783, normalized size = 9.43 \begin{align*} \left [-\frac{\sqrt{c^{2} - d^{2}}{\left (d \cos \left (f x + e\right ) + d\right )} \log \left (\frac{2 \, c d \cos \left (f x + e\right ) -{\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt{c^{2} - d^{2}}{\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) - 2 \,{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}{2 \,{\left ({\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \cos \left (f x + e\right ) +{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f\right )}}, -\frac{\sqrt{-c^{2} + d^{2}}{\left (d \cos \left (f x + e\right ) + d\right )} \arctan \left (-\frac{\sqrt{-c^{2} + d^{2}}{\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) -{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}{{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \cos \left (f x + e\right ) +{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec{\left (e + f x \right )}}{c \sec{\left (e + f x \right )} + c + d \sec ^{2}{\left (e + f x \right )} + d \sec{\left (e + f x \right )}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2391, size = 154, normalized size = 1.86 \begin{align*} \frac{\frac{2 \,{\left (\pi \left \lfloor \frac{f x + e}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\sqrt{-c^{2} + d^{2}}}\right )\right )} d}{{\left (a c - a d\right )} \sqrt{-c^{2} + d^{2}}} + \frac{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a c - a d}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]