Integrand size = 27, antiderivative size = 319 \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {c+d}{c},\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \]
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Time = 0.52 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4014, 4006, 3869, 3917, 4053} \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right )}{a f (c-d)}-\frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} \operatorname {EllipticPi}\left (\frac {c+d}{c},\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right )}{a c f}-\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f (c-d) \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]
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Rule 3869
Rule 3917
Rule 4006
Rule 4014
Rule 4053
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {-a c+a d-a d \sec (e+f x)}{\sqrt {c+d \sec (e+f x)}} \, dx}{a^2 (c-d)}+\frac {a \int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx}{-a c+a d} \\ & = -\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}+\frac {\int \frac {1}{\sqrt {c+d \sec (e+f x)}} \, dx}{a}+\frac {d \int \frac {\sec (e+f x)}{\sqrt {c+d \sec (e+f x)}} \, dx}{a (c-d)} \\ & = \frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {c+d}{c},\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \\ \end{align*}
Time = 8.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=-\frac {4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {d+c \cos (e+f x)}{(c+d) (1+\cos (e+f x))}} \left ((c+d) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+2 (c-2 d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )+4 (-c+d) \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )\right ) \sec ^2(e+f x) \left (\frac {1}{1+\sec (e+f x)}\right )^{3/2}}{a (c-d) f \sqrt {c+d \sec (e+f x)}} \]
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Time = 7.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\left (\cos \left (f x +e \right )+1\right ) \left (2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) c -4 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) d +c \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right )+d \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right )-4 c \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {c -d}{c +d}}\right )+4 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {c -d}{c +d}}\right ) d \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {c +d \sec \left (f x +e \right )}}{a f \left (c -d \right ) \left (d +c \cos \left (f x +e \right )\right )}\) | \(287\) |
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {\int \frac {1}{\sqrt {c + d \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx}{a} \]
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\[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {1}{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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