Integrand size = 39, antiderivative size = 170 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 b g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 (b c-a d) g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d (c+d) f \sqrt {a+b \sec (e+f x)}} \]
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Time = 1.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {4056, 3944, 2886, 2884, 4060} \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 b g \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 g (b c-a d) \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{d f (c+d) \sqrt {a+b \sec (e+f x)}} \]
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Rule 2884
Rule 2886
Rule 3944
Rule 4056
Rule 4060
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx}{d}-\frac {(b c-a d) \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx}{d} \\ & = \frac {\left (b g \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {b+a \cos (e+f x)}} \, dx}{d \sqrt {a+b \sec (e+f x)}}-\frac {\left ((b c-a d) 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}{d \sqrt {a+b \sec (e+f x)}} \\ & = \frac {\left (b g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}} \, dx}{d \sqrt {a+b \sec (e+f x)}}-\frac {\left ((b c-a d) 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}{d \sqrt {a+b \sec (e+f x)}} \\ & = \frac {2 b g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 (b c-a d) 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)}}{d (c+d) f \sqrt {a+b \sec (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 25.56 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.31 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=-\frac {2 i g \sqrt {-\frac {a (-1+\cos (e+f x))}{a+b}} \sqrt {\frac {a (1+\cos (e+f x))}{a-b}} \cot (e+f x) \left (\operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (e+f x)}\right ),\frac {-a+b}{a+b}\right )-\operatorname {EllipticPi}\left (\frac {(a-b) c}{-b c+a d},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (e+f x)}\right ),\frac {-a+b}{a+b}\right )\right ) \sqrt {g \sec (e+f x)} \sqrt {a+b \sec (e+f x)}}{\sqrt {\frac {1}{a-b}} d f \sqrt {b+a \cos (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 8.22 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.62
method | result | size |
default | \(-\frac {2 i g \left (\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) a c d +\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) a \,d^{2}-\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) b c d -\operatorname {EllipticF}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) b \,d^{2}-2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {a -b}{a +b}}\right ) b \,c^{2}+2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -1, i \sqrt {\frac {a -b}{a +b}}\right ) b \,d^{2}-2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -\frac {c -d}{c +d}, i \sqrt {\frac {a -b}{a +b}}\right ) a c d +2 \operatorname {EllipticPi}\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), -\frac {c -d}{c +d}, i \sqrt {\frac {a -b}{a +b}}\right ) b \,c^{2}\right ) \cos \left (f x +e \right ) \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {g \sec \left (f x +e \right )}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}}{f d \left (c +d \right ) \left (c -d \right ) \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(445\) |
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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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\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \sqrt {a + b \sec {\left (e + f x \right )}}}{c + d \sec {\left (e + f x \right )}}\, dx \]
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\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{d \sec \left (f x + e\right ) + c} \,d x } \]
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\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{d \sec \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{c+\frac {d}{\cos \left (e+f\,x\right )}} \,d x \]
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