Integrand size = 39, antiderivative size = 83 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{(c+d) f \sqrt {a+b \sec (e+f x)}} \]
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Time = 0.73 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4060, 2886, 2884} \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{f (c+d) \sqrt {a+b \sec (e+f x)}} \]
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Rule 2884
Rule 2886
Rule 4060
Rubi steps \begin{align*} \text {integral}& = \frac {\left (g \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))} \, dx}{\sqrt {a+b \sec (e+f x)}} \\ & = \frac {\left (g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}} (d+c \cos (e+f x))} \, dx}{\sqrt {a+b \sec (e+f x)}} \\ & = \frac {2 g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{(c+d) f \sqrt {a+b \sec (e+f x)}} \\ \end{align*}
Time = 1.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{(c+d) f \sqrt {a+b \sec (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 7.46 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.69
method | result | size |
default | \(-\frac {2 i g \sqrt {a +b \sec \left (f x +e \right )}\, \cos \left (f x +e \right ) \sqrt {g \sec \left (f x +e \right )}\, \left (2 c \operatorname {EllipticPi}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), -\frac {c -d}{c +d}, i \sqrt {\frac {a -b}{a +b}}\right )-c \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )-d \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}}{f \left (c -d \right ) \left (c +d \right ) \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(223\) |
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Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Timed out} \]
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\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a + b \sec {\left (e + f x \right )}} \left (c + d \sec {\left (e + f x \right )}\right )}\, dx \]
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\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]
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\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \]
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