Integrand size = 19, antiderivative size = 47 \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {\csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{b} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2681, 2653, 2720} \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {\sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{b} \]
[In]
[Out]
Rule 2653
Rule 2681
Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{\sqrt {\sin (a+b x)}} \\ & = \left (\csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = \frac {\csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.55 \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {2 \sqrt [4]{-1} \cos (a+b x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)}}{b \sqrt {\tan (a+b x)}} \]
[In]
[Out]
Time = 0.63 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.30
method | result | size |
default | \(\frac {\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {d \tan \left (b x +a \right )}\, \left (\cot \left (b x +a \right )+\csc \left (b x +a \right )\right ) \sqrt {2}}{b}\) | \(108\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15 \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1)}{b} \]
[In]
[Out]
\[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \sqrt {d \tan {\left (a + b x \right )}} \csc {\left (a + b x \right )}\, dx \]
[In]
[Out]
\[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \csc \left (b x + a\right ) \,d x } \]
[In]
[Out]
\[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \csc \left (b x + a\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \frac {\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sin \left (a+b\,x\right )} \,d x \]
[In]
[Out]