Integrand size = 29, antiderivative size = 141 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c-d} f} \]
[Out]
Time = 0.56 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4023, 4019, 209, 4068} \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]
[In]
[Out]
Rule 209
Rule 4019
Rule 4023
Rule 4068
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx}{a}-\int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{1+a c x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \text {Subst}\left (\int \frac {1}{2+(a c-a d) x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \\ & = \frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c-d} f} \\ \end{align*}
Time = 0.82 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \left (\sqrt {2} \sqrt {c-d} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {d+c \cos (e+f x)}}\right )-\sqrt {c} \arctan \left (\frac {\sqrt {c-d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {d+c \cos (e+f x)}}\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {d+c \cos (e+f x)} \sec (e+f x)}{\sqrt {c} \sqrt {c-d} f \sqrt {a (1+\sec (e+f x))} \sqrt {c+d \sec (e+f x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(114)=228\).
Time = 2.50 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.60
method | result | size |
default | \(-\frac {2 \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {c +d \sec \left (f x +e \right )}\, \left (-\ln \left (\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}-\sqrt {c -d}\, \cot \left (f x +e \right )+\sqrt {c -d}\, \csc \left (f x +e \right )\right ) c^{3}+2 \ln \left (\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}-\sqrt {c -d}\, \cot \left (f x +e \right )+\sqrt {c -d}\, \csc \left (f x +e \right )\right ) c^{2} d -\ln \left (\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}-\sqrt {c -d}\, \cot \left (f x +e \right )+\sqrt {c -d}\, \csc \left (f x +e \right )\right ) c \,d^{2}+\sqrt {2}\, \sqrt {-\left (c -d \right )^{4} c}\, \arctan \left (\frac {\left (c -d \right )^{2} c \sqrt {2}\, \sin \left (f x +e \right )}{\sqrt {-\left (c -d \right )^{4} c}\, \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {c -d}\right ) \cos \left (f x +e \right )}{f a \sqrt {c -d}\, \left (c^{2}-2 c d +d^{2}\right ) c \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}}\) | \(367\) |
[In]
[Out]
none
Time = 0.92 (sec) , antiderivative size = 913, normalized size of antiderivative = 6.48 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\left [\frac {\sqrt {2} a c \sqrt {-\frac {1}{a c - a d}} \log \left (\frac {2 \, \sqrt {2} {\left (c - d\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a c - a d}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (3 \, c - d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (c + d\right )} \cos \left (f x + e\right ) - c + 3 \, d}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 2 \, \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{2 \, a c f}, \frac {\sqrt {2} a c \sqrt {-\frac {1}{a c - a d}} \log \left (\frac {2 \, \sqrt {2} {\left (c - d\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a c - a d}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (3 \, c - d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (c + d\right )} \cos \left (f x + e\right ) - c + 3 \, d}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 4 \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right )}{2 \, a c f}, \frac {\frac {\sqrt {2} a c \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a c - a d} \sin \left (f x + e\right )}\right )}{\sqrt {a c - a d}} - \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a c f}, \frac {\frac {\sqrt {2} a c \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a c - a d} \sin \left (f x + e\right )}\right )}{\sqrt {a c - a d}} - 2 \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right )}{a c f}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {1}{\sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {1}{\sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
[In]
[Out]