Integrand size = 33, antiderivative size = 118 \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {2 \sqrt {\csc (d+e x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \]
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Time = 0.20 (sec) , antiderivative size = 118, 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 = {3247, 3206, 2740} \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {2 \sqrt {\csc (d+e x)} \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {a+b \csc (d+e x)+c \cot (d+e x)}} \]
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Rule 2740
Rule 3206
Rule 3247
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\csc (d+e x)} \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}} \, dx}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \\ & = \frac {\left (\sqrt {\csc (d+e x)} \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}\right ) \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}} \, dx}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \\ & = \frac {2 \sqrt {\csc (d+e x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 1.08 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.87 \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}{b-a \sqrt {1+\frac {c^2}{a^2}}},\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}{b+a \sqrt {1+\frac {c^2}{a^2}}}\right ) \sqrt {\csc (d+e x)} \sec \left (d+e x+\arctan \left (\frac {c}{a}\right )\right ) \sqrt {b+c \cos (d+e x)+a \sin (d+e x)} \sqrt {-\frac {a \sqrt {1+\frac {c^2}{a^2}} \left (-1+\sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )\right )}{b+a \sqrt {1+\frac {c^2}{a^2}}}} \sqrt {\frac {a \sqrt {1+\frac {c^2}{a^2}} \left (1+\sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )\right )}{-b+a \sqrt {1+\frac {c^2}{a^2}}}} \sqrt {b+a \sqrt {1+\frac {c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}}{a \sqrt {1+\frac {c^2}{a^2}} e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \]
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Result contains complex when optimal does not.
Time = 31.07 (sec) , antiderivative size = 644, normalized size of antiderivative = 5.46
method | result | size |
default | \(\frac {2 i \sqrt {2}\, \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right ) \sqrt {\csc \left (e x +d \right )}\, \sqrt {a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {-\frac {\left (i \cos \left (e x +d \right )+i-\sin \left (e x +d \right )\right )^{2} \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (1+\cos \left (e x +d \right )\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}}{2}, \sqrt {\frac {\left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\right ) \sqrt {-\frac {\left (i \cos \left (e x +d \right )+i-\sin \left (e x +d \right )\right )^{2} \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (1+\cos \left (e x +d \right )\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \sqrt {\frac {i \left (\sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )-b \cos \left (e x +d \right )+c \cos \left (e x +d \right )+a \sin \left (e x +d \right )+b -c \right )}{\left (i \sin \left (e x +d \right )-\cos \left (e x +d \right )+1\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \sqrt {-\frac {i \left (\sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )+b \cos \left (e x +d \right )-c \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b +c \right )}{\left (i \sin \left (e x +d \right )-\cos \left (e x +d \right )+1\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \left (i \sin \left (e x +d \right )^{2}-\cos \left (e x +d \right ) \sin \left (e x +d \right )\right )}{e \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}\) | \(644\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 508, normalized size of antiderivative = 4.31 \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {{\left (i \, a - c\right )} \sqrt {-2 i \, a - 2 \, c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} + 6 i \, a c^{3} - 3 \, c^{4} + 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (-9 i \, a^{5} b + 8 i \, a^{3} b^{3} + 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} + 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {-2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (e x + d\right ) - 3 \, {\left (i \, a^{2} + i \, c^{2}\right )} \sin \left (e x + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right ) + \sqrt {2 i \, a - 2 \, c} {\left (-i \, a - c\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} - 6 i \, a c^{3} - 3 \, c^{4} - 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (9 i \, a^{5} b - 8 i \, a^{3} b^{3} - 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} - 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (e x + d\right ) - 3 \, {\left (-i \, a^{2} - i \, c^{2}\right )} \sin \left (e x + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right )}{{\left (a^{2} + c^{2}\right )} e} \]
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\[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\int \frac {\sqrt {\csc {\left (d + e x \right )}}}{\sqrt {a + b \csc {\left (d + e x \right )} + c \cot {\left (d + e x \right )}}}\, dx \]
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\[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\int { \frac {\sqrt {\csc \left (e x + d\right )}}{\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a}} \,d x } \]
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\[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\int { \frac {\sqrt {\csc \left (e x + d\right )}}{\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\int \frac {\sqrt {\frac {1}{\sin \left (d+e\,x\right )}}}{\sqrt {a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}}} \,d x \]
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