Integrand size = 33, antiderivative size = 95 \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {4053} \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}} \]
[In]
[Out]
Rule 4053
Rubi steps \begin{align*} \text {integral}& = \frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(264\) vs. \(2(95)=190\).
Time = 6.44 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.78 \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {1+\sec (e+f x)}}{\left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}}}+\frac {\sec ^5\left (\frac {1}{2} (e+f x)\right ) \sqrt {1+\sec (e+f x)} \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {3}{2} (e+f x)\right )\right )}{\left (\frac {1}{1+\cos (e+f x)}\right )^{3/2}}-8 \sqrt {\sec (e+f x)} \left (\sin (e+f x)-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{4 c f (1+\sec (e+f x))} \]
[In]
[Out]
Time = 7.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.29
method | result | size |
default | \(\frac {\left (-a -b \right ) \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {a +b \sec \left (f x +e \right )}}{c f \left (b +a \cos \left (f x +e \right )\right )}\) | \(123\) |
[In]
[Out]
\[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{c \sec \left (f x + e\right ) + c} \,d x } \]
[In]
[Out]
\[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\frac {\int \frac {\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{c} \]
[In]
[Out]
\[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{c \sec \left (f x + e\right ) + c} \,d x } \]
[In]
[Out]
\[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{c \sec \left (f x + e\right ) + c} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}}{\cos \left (e+f\,x\right )\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
[In]
[Out]