Optimal. Leaf size=61 \[ \frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{4}{5}\right )}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0568292, antiderivative size = 61, 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 = {3858, 2661} \[ \frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right )}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3858
Rule 2661
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{2+3 \sec (c+d x)}} \, dx &=\frac{\left (\sqrt{3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{3+2 \cos (c+d x)}} \, dx}{\sqrt{2+3 \sec (c+d x)}}\\ &=\frac{2 \sqrt{3+2 \cos (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{4}{5}\right ) \sqrt{\sec (c+d x)}}{\sqrt{5} d \sqrt{2+3 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0552023, size = 61, normalized size = 1. \[ \frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{4}{5}\right )}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.217, size = 142, normalized size = 2.3 \begin{align*}{\frac{-{\frac{i}{5}}\sqrt{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{10}\sqrt{2}}{d \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) -3 \right ) }{\it EllipticF} \left ({\frac{{\frac{i}{5}} \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},\sqrt{5} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \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{\sec \left (d x + c\right )}}{\sqrt{3 \, \sec \left (d x + c\right ) + 2}}\,{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{\sec \left (d x + c\right )}}{\sqrt{3 \, \sec \left (d x + c\right ) + 2}}, 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{\sec{\left (c + d x \right )}}}{\sqrt{3 \sec{\left (c + d x \right )} + 2}}\, 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{\sec \left (d x + c\right )}}{\sqrt{3 \, \sec \left (d x + c\right ) + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]