Integrand size = 25, antiderivative size = 60 \[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2894} \[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right )}{\sqrt {5} d} \]
[In]
[Out]
Rule 2894
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {-\tan ^2(c+d x)}}{\sqrt {5} d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(60)=120\).
Time = 1.44 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3-2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}}{\sqrt {3}}\right ),6\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (54 ) = 108\).
Time = 7.54 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {\left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )}{d \left (-3+2 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\) | \(113\) |
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {3 - 2 \cos {\left (c + d x \right )}} \sqrt {\cos {\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {3-2\,\cos \left (c+d\,x\right )}} \,d x \]
[In]
[Out]