Optimal. Leaf size=108 \[ \frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),-4\right )}{d \sqrt{2-3 \sec (c+d x)}}+\frac{\sqrt{2-3 \sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.198349, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3862, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{2-3 \sec (c+d x)}}+\frac{\sqrt{2-3 \sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |-4\right )}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3862
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-3 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx &=\frac{1}{2} \int \frac{\sqrt{2-3 \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx+\frac{3}{2} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{2-3 \sec (c+d x)}} \, dx\\ &=\frac{\sqrt{2-3 \sec (c+d x)} \int \sqrt{-3+2 \cos (c+d x)} \, dx}{2 \sqrt{-3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (3 \sqrt{-3+2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{-3+2 \cos (c+d x)}} \, dx}{2 \sqrt{2-3 \sec (c+d x)}}\\ &=\frac{\sqrt{2-3 \sec (c+d x)} \int \sqrt{3-2 \cos (c+d x)} \, dx}{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{3-2 \cos (c+d x)}} \, dx}{2 \sqrt{2-3 \sec (c+d x)}}\\ &=\frac{E\left (\left .\frac{1}{2} (c+d x)\right |-4\right ) \sqrt{2-3 \sec (c+d x)}}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{3 \sqrt{3-2 \cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |-4\right ) \sqrt{\sec (c+d x)}}{d \sqrt{2-3 \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0779803, size = 68, normalized size = 0.63 \[ -\frac{\sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \left (E\left (\left .\frac{1}{2} (c+d x)\right |-4\right )-3 \text{EllipticF}\left (\frac{1}{2} (c+d x),-4\right )\right )}{d \sqrt{2-3 \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.28, size = 405, normalized size = 3.8 \begin{align*} -{\frac{1}{10\,d\sin \left ( dx+c \right ) \left ( -3+2\,\cos \left ( dx+c \right ) \right ) } \left ( 2\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{5}{\it EllipticF} \left ({\frac{i\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{2}-5\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{5}{\it EllipticE} \left ({\frac{i\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{2}+2\,i\sqrt{5}{\it EllipticF} \left ({\frac{i\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) -5\,i\sqrt{5}{\it EllipticE} \left ({\frac{i\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) \sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) +20\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-50\,\cos \left ( dx+c \right ) +30 \right ) \sqrt{{\frac{-3+2\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \right ) \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{1}{\sqrt{-3 \, \sec \left (d x + c\right ) + 2} \sqrt{\sec \left (d x + c\right )}}\,{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{-3 \, \sec \left (d x + c\right ) + 2} \sqrt{\sec \left (d x + c\right )}}{3 \, \sec \left (d x + c\right )^{2} - 2 \, \sec \left (d x + c\right )}, 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{1}{\sqrt{2 - 3 \sec{\left (c + d x \right )}} \sqrt{\sec{\left (c + d x \right )}}}\, 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{1}{\sqrt{-3 \, \sec \left (d x + c\right ) + 2} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]