Integrand size = 33, antiderivative size = 154 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 c \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (a+b) f \sqrt {c+d \sin (e+f x)}} \]
[Out]
Time = 0.33 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3014, 2886, 2884} \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 c \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f (a+b) \sqrt {c+d \sin (e+f x)}} \]
[In]
[Out]
Rule 2884
Rule 2886
Rule 3014
Rubi steps \begin{align*} \text {integral}& = \frac {c \int \frac {\csc (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{a}+\frac {(-b c+a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{a} \\ & = \frac {\left (c \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {\csc (e+f x)}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{a \sqrt {c+d \sin (e+f x)}}+\frac {\left ((-b c+a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{a \sqrt {c+d \sin (e+f x)}} \\ & = \frac {2 c \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (a+b) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 30.02 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.16 \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 i \left (\operatorname {EllipticPi}\left (\frac {c+d}{c},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-a d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{a \sqrt {-\frac {1}{c+d}} f} \]
[In]
[Out]
Time = 1.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.23
method | result | size |
default | \(-\frac {2 \left (\Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {c -d}{c}, \sqrt {\frac {c -d}{c +d}}\right )-\Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, -\frac {\left (c -d \right ) b}{a d -b c}, \sqrt {\frac {c -d}{c +d}}\right )\right ) \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \left (c -d \right )}{a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(190\) |
[In]
[Out]
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\left (a + b \sin {\left (e + f x \right )}\right ) \sin {\left (e + f x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sin \left (f x + e\right )} \,d x } \]
[In]
[Out]
\[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sin \left (f x + e\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\csc (e+f x) \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]
[In]
[Out]