Integrand size = 33, antiderivative size = 209 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=-\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) c f}+\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{(a-b) c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \]
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Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4057, 3917, 4053} \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac {2 \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{c f (a-b)} \]
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Rule 3917
Rule 4053
Rule 4057
Rubi steps \begin{align*} \text {integral}& = -\frac {b \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{(a-b) c}-\frac {c \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx}{-a c+b c} \\ & = -\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) c f}+\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)}}{(a-b) c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \\ \end{align*}
Time = 13.34 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.79 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (b+a \cos (e+f x)) \sec ^2(e+f x) \left (\frac {2 \sin (e+f x)}{-a+b}-\frac {2 \tan \left (\frac {1}{2} (e+f x)\right )}{-a+b}\right )}{f \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))}-\frac {2 \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^{\frac {3}{2}}(e+f x) \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)} \left ((a-b) E\left (\arcsin \left (\sqrt {\frac {a-b}{a+b}} \tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {(b+a \cos (e+f x)) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+\sqrt {2} \sqrt {\frac {a-b}{a+b}} \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} (b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{\left (\frac {a-b}{a+b}\right )^{3/2} (a+b) f \sqrt {\cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )} \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \]
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Time = 5.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\left (\cos \left (f x +e \right )+1\right ) \left (2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b -a \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )-b \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {a +b \sec \left (f x +e \right )}}{c f \left (a -b \right ) \left (b +a \cos \left (f x +e \right )\right )}\) | \(191\) |
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\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {a + b \sec {\left (e + f x \right )}}}\, dx}{c} \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
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