Integrand size = 19, antiderivative size = 58 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2702, 335, 304, 209, 212} \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f} \]
[In]
[Out]
Rule 209
Rule 212
Rule 304
Rule 335
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {x}}{-1+\frac {x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{b f} \\ & = \frac {2 \text {Subst}\left (\int \frac {x^2}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{b f} \\ & = -\frac {b \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{f}+\frac {b \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{f} \\ & = \frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (2 \arctan \left (\sqrt {\sec (e+f x)}\right )+\log \left (1-\sqrt {\sec (e+f x)}\right )-\log \left (1+\sqrt {\sec (e+f x)}\right )\right ) \sqrt {b \sec (e+f x)}}{2 f \sqrt {\sec (e+f x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(46)=92\).
Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.60
method | result | size |
default | \(\frac {\sqrt {b \sec \left (f x +e \right )}\, \left (\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )-\ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right )\right ) \cos \left (f x +e \right )}{2 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(151\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (46) = 92\).
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 4.26 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\left [\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, f}, -\frac {2 \, \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) - \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, f}\right ] \]
[In]
[Out]
\[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}} \csc {\left (e + f x \right )}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.24 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {b {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{\sqrt {b}} + \frac {\log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{\sqrt {b}}\right )}}{2 \, f} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {b^{2} {\left (\frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{f} \]
[In]
[Out]
Timed out. \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{\sin \left (e+f\,x\right )} \,d x \]
[In]
[Out]