Integrand size = 25, antiderivative size = 109 \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (c+d x)}{a d} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2895} \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d} \]
[In]
[Out]
Rule 2895
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (c+d x)}{a d} \\ \end{align*}
Time = 3.80 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\frac {8 a \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {a+b \sin (c+d x)}{a (-1+\sin (c+d x))}}\right ),\frac {2 a}{a-b}\right ) \sec (c+d x) \sqrt {-\frac {(a+b) \sin (c+d x) (a+b \sin (c+d x))}{a^2 (-1+\sin (c+d x))^2}} \sin ^4\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{(a+b) d \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(101)=202\).
Time = 4.03 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.42
method | result | size |
default | \(-\frac {\sqrt {\frac {-\cot \left (d x +c \right ) a +\csc \left (d x +c \right ) a +\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {\sqrt {-a^{2}+b^{2}}+\cot \left (d x +c \right ) a -b -\csc \left (d x +c \right ) a}{\sqrt {-a^{2}+b^{2}}}}\, \sqrt {\frac {a \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{b +\sqrt {-a^{2}+b^{2}}}}\, F\left (\sqrt {\frac {-\cot \left (d x +c \right ) a +\csc \left (d x +c \right ) a +\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \left (\sin ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}\, \left (b +\sqrt {-a^{2}+b^{2}}\right )}{d \sqrt {a +b \sin \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) a}\) | \(264\) |
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \sin {\left (c + d x \right )}} \sqrt {\sin {\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {1}{\sqrt {\sin \left (c+d\,x\right )}\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]
[In]
[Out]