Optimal. Leaf size=62 \[ \frac{2 a (3 A+B) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0944036, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {4001, 3792} \[ \frac{2 a (3 A+B) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx &=\frac{2 B \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{3} (3 A+B) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (3 A+B) \tan (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{2 B \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.158314, size = 53, normalized size = 0.85 \[ \frac{2 \tan (c+d x) \sqrt{a (\sec (c+d x)+1)} ((3 A+2 B) \cos (c+d x)+B)}{3 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.273, size = 70, normalized size = 1.1 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 3\,A\cos \left ( dx+c \right ) +2\,B\cos \left ( dx+c \right ) +B \right ) }{3\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.462124, size = 169, normalized size = 2.73 \begin{align*} \frac{2 \,{\left ({\left (3 \, A + 2 \, B\right )} \cos \left (d x + c\right ) + B\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\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*} \int \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \left (A + B \sec{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 4.60645, size = 174, normalized size = 2.81 \begin{align*} -\frac{2 \,{\left (3 \, \sqrt{2} A a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, \sqrt{2} B a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (3 \, \sqrt{2} A a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + \sqrt{2} B a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{3 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]