Integrand size = 22, antiderivative size = 407 \[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\frac {2 x^{3/2}}{3 a}+\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3} \]
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Time = 1.05 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4290, 4276, 3404, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d \sqrt {b^2-a^2}}-\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d \sqrt {b^2-a^2}}+\frac {2 x^{3/2}}{3 a} \]
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3404
Rule 4276
Rule 4290
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2}{a+b \csc (c+d x)} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^2}{a}-\frac {b x^2}{a (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{3/2}}{3 a}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {2 x^{3/2}}{3 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {2 x^{3/2}}{3 a}+\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}-\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}} \\ & = \frac {2 x^{3/2}}{3 a}+\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {(4 i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}+\frac {(4 i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d} \\ & = \frac {2 x^{3/2}}{3 a}+\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {(4 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(4 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2} \\ & = \frac {2 x^{3/2}}{3 a}+\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(4 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(4 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^3} \\ & = \frac {2 x^{3/2}}{3 a}+\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {4 b \sqrt {x} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3} \\ \end{align*}
Time = 3.48 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\frac {2 \left (\sqrt {a^2-b^2} d^3 x^{3/2}-3 b d^2 x \log \left (1-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+3 b d^2 x \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )+6 i b d \sqrt {x} \operatorname {PolyLog}\left (2,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )-6 i b d \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )-6 b \operatorname {PolyLog}\left (3,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-i b+\sqrt {a^2-b^2}}\right )+6 b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right )\right )}{3 a \sqrt {a^2-b^2} d^3} \]
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\[\int \frac {\sqrt {x}}{a +b \csc \left (c +d \sqrt {x}\right )}d x\]
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\[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {\sqrt {x}}{b \csc \left (d \sqrt {x} + c\right ) + a} \,d x } \]
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\[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int \frac {\sqrt {x}}{a + b \csc {\left (c + d \sqrt {x} \right )}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {\sqrt {x}}{b \csc \left (d \sqrt {x} + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {x}}{a+b \csc \left (c+d \sqrt {x}\right )} \, dx=\int \frac {\sqrt {x}}{a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}} \,d x \]
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