Integrand size = 25, antiderivative size = 142 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=-\frac {2 b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \]
[Out]
Time = 0.41 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4349, 3947, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {2 b \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \]
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3941
Rule 3943
Rule 3947
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{a}-\frac {\left (b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{a} \\ & = -\frac {\left (b \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{a \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{a \sqrt {b+a \cos (c+d x)}} \\ & = -\frac {\left (b \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{a \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{a \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = -\frac {2 b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.94 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)} \left (i (a+b) \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )-i a \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+\sqrt {\frac {1}{1+\cos (c+d x)}} (b+a \cos (c+d x)) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d \sqrt {a+b \sec (c+d x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(738\) vs. \(2(188)=376\).
Time = 7.38 (sec) , antiderivative size = 739, normalized size of antiderivative = 5.20
method | result | size |
default | \(\frac {2 \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}}\, \sqrt {\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (\sqrt {\frac {a -b}{a +b}}\, a \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-\sqrt {\frac {a -b}{a +b}}\, b \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\sqrt {-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{a +b}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) a -\sqrt {-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{a +b}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) a +\sqrt {-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{a +b}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}\, \operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) b -\sqrt {\frac {a -b}{a +b}}\, a \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-\sqrt {\frac {a -b}{a +b}}\, b \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )}{d a \sqrt {\frac {a -b}{a +b}}\, \left (a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b \right )}\) | \(739\) |
risch | \(-\frac {i \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{a d \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {i \left (-\frac {2 \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}}+\frac {2 \left (b +\sqrt {-a^{2}+b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right ) a}{b +\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {-b +\sqrt {-a^{2}+b^{2}}}{a}}{-\frac {b +\sqrt {-a^{2}+b^{2}}}{a}-\frac {-b +\sqrt {-a^{2}+b^{2}}}{a}}}\, \sqrt {-\frac {{\mathrm e}^{i \left (d x +c \right )} a}{b +\sqrt {-a^{2}+b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-a^{2}+b^{2}}}{a}-\frac {-b +\sqrt {-a^{2}+b^{2}}}{a}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right ) a}{b +\sqrt {-a^{2}+b^{2}}}}, \sqrt {-\frac {b +\sqrt {-a^{2}+b^{2}}}{a \left (-\frac {b +\sqrt {-a^{2}+b^{2}}}{a}-\frac {-b +\sqrt {-a^{2}+b^{2}}}{a}\right )}}\right )+\frac {\left (-b +\sqrt {-a^{2}+b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +\sqrt {-a^{2}+b^{2}}}{a}\right ) a}{b +\sqrt {-a^{2}+b^{2}}}}, \sqrt {-\frac {b +\sqrt {-a^{2}+b^{2}}}{a \left (-\frac {b +\sqrt {-a^{2}+b^{2}}}{a}-\frac {-b +\sqrt {-a^{2}+b^{2}}}{a}\right )}}\right )}{a}\right )}{a \sqrt {a \,{\mathrm e}^{3 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )} a}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}}{d \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(819\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.50 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 i \, \sqrt {2} \sqrt {a} b {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 2 i \, \sqrt {2} \sqrt {a} b {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + 3 i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right )}{3 \, a^{2} d} \]
[In]
[Out]
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {a + b \sec {\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {b \sec \left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {b \sec \left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}} \,d x \]
[In]
[Out]