Integrand size = 25, antiderivative size = 111 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{b f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{b f \sqrt {a+b \sin ^2(e+f x)}} \]
(cos(f*x+e)^2)^(1/2)/cos(f*x+e)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*(a+b*si n(f*x+e)^2)^(1/2)/b/f/(1+b*sin(f*x+e)^2/a)^(1/2)-a*(cos(f*x+e)^2)^(1/2)/co s(f*x+e)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*(1+b*sin(f*x+e)^2/a)^(1/2)/b/f /(a+b*sin(f*x+e)^2)^(1/2)
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {\sqrt {2 a+b-b \cos (2 (e+f x))} \left (E\left (e+f x\left |-\frac {b}{a}\right .\right )-\operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )}{b f \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}}} \]
(Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*(EllipticE[e + f*x, -(b/a)] - Elliptic F[e + f*x, -(b/a)]))/(b*f*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a])
Time = 0.61 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3651, 3042, 3657, 3042, 3656, 3662, 3042, 3661}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (e+f x)^2}{\sqrt {a+b \sin (e+f x)^2}}dx\) |
\(\Big \downarrow \) 3651 |
\(\displaystyle \frac {\int \sqrt {b \sin ^2(e+f x)+a}dx}{b}-\frac {a \int \frac {1}{\sqrt {b \sin ^2(e+f x)+a}}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {b \sin (e+f x)^2+a}dx}{b}-\frac {a \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\) |
\(\Big \downarrow \) 3657 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}dx}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin (e+f x)^2}{a}+1}dx}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\) |
\(\Big \downarrow \) 3662 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}dx}{b \sqrt {a+b \sin ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin (e+f x)^2}{a}+1}}dx}{b \sqrt {a+b \sin ^2(e+f x)}}\) |
\(\Big \downarrow \) 3661 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{b f \sqrt {a+b \sin ^2(e+f x)}}\) |
(EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*f*Sqrt[1 + (b*S in[e + f*x]^2)/a]) - (a*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x ]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])
3.2.49.3.1 Defintions of rubi rules used
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]^2], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre eQ[{a, b, e, f, A, B}, x]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a ]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)] Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[ 1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a + b*Sin[e + f*x]^2] Int[1/Sqrt[1 + (b*Si n[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Time = 1.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \left (F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )-E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )\right )}{b \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(93\) |
-a*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)/b*(EllipticF(sin(f*x+ e),(-1/a*b)^(1/2))-EllipticE(sin(f*x+e),(-1/a*b)^(1/2)))/cos(f*x+e)/(a+b*s in(f*x+e)^2)^(1/2)/f
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]