Optimal. Leaf size=92 \[ \frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.164096, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2829, 3968} \[ \frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2829
Rule 3968
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sec (c+d x)}}{1+\cos (c+d x)} \, dx &=\int \frac{\sec (c+d x) \sqrt{a+b \sec (c+d x)}}{1+\sec (c+d x)} \, dx\\ &=\frac{E\left (\sin ^{-1}\left (\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )|\frac{a-b}{a+b}\right ) \sqrt{\frac{1}{1+\sec (c+d x)}} \sqrt{a+b \sec (c+d x)}}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (1+\sec (c+d x))}}}\\ \end{align*}
Mathematica [A] time = 1.56502, size = 85, normalized size = 0.92 \[ \frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )}{d \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.305, size = 150, normalized size = 1.6 \begin{align*} -{\frac{ \left ( -a-b \right ) \left ( \cos \left ( dx+c \right ) -1 \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}}{d \left ( a\cos \left ( dx+c \right ) +b \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{\it EllipticE} \left ({\frac{\cos \left ( dx+c \right ) -1}{\sin \left ( dx+c \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) \sqrt{{\frac{a\cos \left ( dx+c \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{a\cos \left ( dx+c \right ) +b}{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}, x\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 \frac{\sqrt{a + b \sec{\left (c + d x \right )}}}{\cos{\left (c + d x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (d x + c\right ) + a}}{\cos \left (d x + c\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]