Integrand size = 25, antiderivative size = 129 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\frac {4 \sqrt {2-3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-3+2 \sec (c+d x)}}-\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {-3+2 \sec (c+d x)}}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3947, 3941, 2733, 3943, 2741} \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\frac {4 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {6}{5}\right )}{3 \sqrt {5} d \sqrt {2 \sec (c+d x)-3}}-\frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)-3} E\left (\frac {1}{2} (c+d x+\pi )|\frac {6}{5}\right )}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]
[In]
[Out]
Rule 2733
Rule 2741
Rule 3941
Rule 3943
Rule 3947
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \int \frac {\sqrt {-3+2 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx\right )+\frac {2}{3} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3+2 \sec (c+d x)}} \, dx \\ & = \frac {\left (2 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {2-3 \cos (c+d x)}} \, dx}{3 \sqrt {-3+2 \sec (c+d x)}}-\frac {\sqrt {-3+2 \sec (c+d x)} \int \sqrt {2-3 \cos (c+d x)} \, dx}{3 \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {4 \sqrt {2-3 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-3+2 \sec (c+d x)}}-\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+\pi +d x)|\frac {6}{5}\right ) \sqrt {-3+2 \sec (c+d x)}}{3 d \sqrt {2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\frac {\sqrt {-2+3 \cos (c+d x)} \left (2 E\left (\left .\frac {1}{2} (c+d x)\right |6\right )+4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),6\right )\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {-3+2 \sec (c+d x)}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 7.57 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.57
method | result | size |
default | \(-\frac {2 \sqrt {-\frac {5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (3 i \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i \sqrt {5}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \sqrt {-5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}-i \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i \sqrt {5}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \sqrt {-5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}+5 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}{3 d \left (5 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \sqrt {-\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(332\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right ) \sqrt {2}}{3 d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (\frac {-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+\frac {8 \,{\mathrm e}^{i \left (d x +c \right )}}{3}-2}{\sqrt {\left (-3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}-3\right ) {\mathrm e}^{i \left (d x +c \right )}}}+\frac {\left (-\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}+\frac {i \sqrt {5}}{3}}{-\frac {2}{3}+\frac {i \sqrt {5}}{3}}}\, \sqrt {30}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{\frac {2}{3}-\frac {i \sqrt {5}}{3}}}\, \left (-\frac {2 i \sqrt {5}\, \operatorname {EllipticE}\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}+\frac {i \sqrt {5}}{3}}{-\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )}{3}+\left (\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {2}{3}+\frac {i \sqrt {5}}{3}}{-\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )\right )}{5 \sqrt {-3 \,{\mathrm e}^{3 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {-{\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}}{d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(576\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=-\frac {4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + 4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right )}{27 \, d} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 \sec {\left (c + d x \right )} - 3} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}-3}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
[In]
[Out]