Integrand size = 22, antiderivative size = 66 \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\frac {2 \sqrt {x}}{a}+\frac {4 b \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4290, 3868, 2739, 632, 212} \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\frac {4 b \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}+\frac {2 \sqrt {x}}{a} \]
[In]
[Out]
Rule 212
Rule 632
Rule 2739
Rule 3868
Rule 4290
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{a+b \csc (c+d x)} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \sqrt {x}}{a}-\frac {2 \text {Subst}\left (\int \frac {1}{1+\frac {a \sin (c+d x)}{b}} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {2 \sqrt {x}}{a}-\frac {4 \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a d} \\ & = \frac {2 \sqrt {x}}{a}+\frac {8 \text {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a d} \\ & = \frac {2 \sqrt {x}}{a}+\frac {4 b \text {arctanh}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\frac {2 \left (\frac {c}{d}+\sqrt {x}-\frac {2 b \arctan \left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2} d}\right )}{a} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a}-\frac {4 b \arctan \left (\frac {2 b \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a \sqrt {-a^{2}+b^{2}}}}{d}\) | \(73\) |
default | \(\frac {\frac {4 \arctan \left (\tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{a}-\frac {4 b \arctan \left (\frac {2 b \tan \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a \sqrt {-a^{2}+b^{2}}}}{d}\) | \(73\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 275, normalized size of antiderivative = 4.17 \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\left [\frac {2 \, {\left (a^{2} - b^{2}\right )} d \sqrt {x} + \sqrt {a^{2} - b^{2}} b \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d \sqrt {x} + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} a \cos \left (d \sqrt {x} + c\right ) + a^{2} + b^{2} + 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d \sqrt {x} + c\right ) + a b\right )} \sin \left (d \sqrt {x} + c\right )}{a^{2} \cos \left (d \sqrt {x} + c\right )^{2} - 2 \, a b \sin \left (d \sqrt {x} + c\right ) - a^{2} - b^{2}}\right )}{{\left (a^{3} - a b^{2}\right )} d}, \frac {2 \, {\left ({\left (a^{2} - b^{2}\right )} d \sqrt {x} + \sqrt {-a^{2} + b^{2}} b \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \sin \left (d \sqrt {x} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \cos \left (d \sqrt {x} + c\right )}\right )\right )}}{{\left (a^{3} - a b^{2}\right )} d}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{\sqrt {x} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=-\frac {4 \, {\left (\pi \left \lfloor \frac {d \sqrt {x} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} b}{\sqrt {-a^{2} + b^{2}} a d} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a d} \]
[In]
[Out]
Time = 19.52 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\frac {2\,\sqrt {x}}{a}-\frac {2\,b\,\ln \left (b\,{\mathrm {e}}^{d\,\sqrt {x}\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,2{}\mathrm {i}-\frac {2\,b\,\left (a\,1{}\mathrm {i}+b\,{\mathrm {e}}^{d\,\sqrt {x}\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )}{\sqrt {a+b}\,\sqrt {a-b}}\right )}{a\,d\,\sqrt {a+b}\,\sqrt {a-b}}+\frac {2\,b\,\ln \left (b\,{\mathrm {e}}^{d\,\sqrt {x}\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,2{}\mathrm {i}+\frac {2\,b\,\left (a\,1{}\mathrm {i}+b\,{\mathrm {e}}^{d\,\sqrt {x}\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )}{\sqrt {a+b}\,\sqrt {a-b}}\right )}{a\,d\,\sqrt {a+b}\,\sqrt {a-b}} \]
[In]
[Out]