Integrand size = 39, antiderivative size = 168 \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a f \sqrt {c+d \sec (e+f x)}}+\frac {2 (a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a (a+b) f \sqrt {c+d \sec (e+f x)}} \]
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Time = 1.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3041, 4056, 3944, 2886, 2884, 4060} \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 (a c-b d) \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f (a+b) \sqrt {c+d \sec (e+f x)}}+\frac {2 d \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right )}{a f \sqrt {c+d \sec (e+f x)}} \]
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Rule 2884
Rule 2886
Rule 3041
Rule 3944
Rule 4056
Rule 4060
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(g \sec (e+f x))^{3/2} \sqrt {c+d \sec (e+f x)}}{b+a \sec (e+f x)} \, dx}{g} \\ & = \frac {d \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {c+d \sec (e+f x)}} \, dx}{a g}+\frac {(a c-b d) \int \frac {(g \sec (e+f x))^{3/2}}{(b+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx}{a g} \\ & = \frac {\left (d \sqrt {d+c \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {d+c \cos (e+f x)}} \, dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {\left ((a c-b d) \sqrt {d+c \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(a+b \cos (e+f x)) \sqrt {d+c \cos (e+f x)}} \, dx}{a \sqrt {c+d \sec (e+f x)}} \\ & = \frac {\left (d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}} \, dx}{a \sqrt {c+d \sec (e+f x)}}+\frac {\left ((a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(a+b \cos (e+f x)) \sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}} \, dx}{a \sqrt {c+d \sec (e+f x)}} \\ & = \frac {2 d \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a f \sqrt {c+d \sec (e+f x)}}+\frac {2 (a c-b d) \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (e+f x),\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{a (a+b) f \sqrt {c+d \sec (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 25.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=-\frac {2 i \sqrt {-\frac {c (-1+\cos (e+f x))}{c+d}} \sqrt {\frac {c (1+\cos (e+f x))}{c-d}} \cot (e+f x) \left (\operatorname {EllipticPi}\left (1-\frac {c}{d},i \text {arcsinh}\left (\sqrt {\frac {1}{c-d}} \sqrt {d+c \cos (e+f x)}\right ),\frac {-c+d}{c+d}\right )-\operatorname {EllipticPi}\left (\frac {b (-c+d)}{-a c+b d},i \text {arcsinh}\left (\sqrt {\frac {1}{c-d}} \sqrt {d+c \cos (e+f x)}\right ),\frac {-c+d}{c+d}\right )\right ) \sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a \sqrt {\frac {1}{c-d}} f \sqrt {d+c \cos (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 8.24 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.63
method | result | size |
default | \(\frac {2 i \sqrt {g \sec \left (f x +e \right )}\, \cos \left (f x +e \right ) \sqrt {c +d \sec \left (f x +e \right )}\, \left (\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a^{2} c -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a^{2} d +\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a b c -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {c -d}{c +d}}\right ) a b d +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {c -d}{c +d}}\right ) a^{2} d -2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {c -d}{c +d}}\right ) b^{2} d -2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right ) a b c +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right ) b^{2} d \right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}}{f a \left (a -b \right ) \left (a +b \right ) \left (d +c \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(442\) |
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Timed out. \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int \frac {\sqrt {g \sec {\left (e + f x \right )}} \sqrt {c + d \sec {\left (e + f x \right )}}}{a + b \cos {\left (e + f x \right )}}\, dx \]
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\[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sqrt {g \sec \left (f x + e\right )}}{b \cos \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int { \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sqrt {g \sec \left (f x + e\right )}}{b \cos \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {g \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\int \frac {\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {g}{\cos \left (e+f\,x\right )}}}{a+b\,\cos \left (e+f\,x\right )} \,d x \]
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