Integrand size = 25, antiderivative size = 221 \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(a-b) \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a b (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]
-1/3*cos(f*x+e)*sin(f*x+e)/(a+b)/f/(a+b*sin(f*x+e)^2)^(3/2)-1/3*(a-b)*cos( f*x+e)*sin(f*x+e)/a/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(a-b)*(cos(f*x+ e)^2)^(1/2)/cos(f*x+e)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*(a+b*sin(f*x+e)^ 2)^(1/2)/a/b/(a+b)^2/f/(1+b*sin(f*x+e)^2/a)^(1/2)+1/3*(cos(f*x+e)^2)^(1/2) /cos(f*x+e)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*(1+b*sin(f*x+e)^2/a)^(1/2)/ b/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)
Time = 1.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 a^2 (a-b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (4 a^2+a b-b^2+b (-a+b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{6 a b (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
(-2*a^2*(a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x , -(b/a)] + 2*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*Ellipti cF[e + f*x, -(b/a)] - Sqrt[2]*b*(4*a^2 + a*b - b^2 + b*(-a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*a*b*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)] )^(3/2))
Time = 1.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3652, 3042, 3652, 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.
| \(\displaystyle \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (e+f x)^2}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\int \frac {a \sin ^2(e+f x)+a}{\left (b \sin ^2(e+f x)+a\right )^{3/2}}dx}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a \sin (e+f x)^2+a}{\left (b \sin (e+f x)^2+a\right )^{3/2}}dx}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\int \frac {2 a^2-a (a-b) \sin ^2(e+f x)}{\sqrt {b \sin ^2(e+f x)+a}}dx}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 a^2-a (a-b) \sin (e+f x)^2}{\sqrt {b \sin (e+f x)^2+a}}dx}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \int \frac {1}{\sqrt {b \sin ^2(e+f x)+a}}dx}{b}-\frac {a (a-b) \int \sqrt {b \sin ^2(e+f x)+a}dx}{b}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}-\frac {a (a-b) \int \sqrt {b \sin (e+f x)^2+a}dx}{b}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3657 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}-\frac {a (a-b) \sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}dx}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}-\frac {a (a-b) \sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin (e+f x)^2}{a}+1}dx}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}-\frac {a (a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3662 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}dx}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {a (a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin (e+f x)^2}{a}+1}}dx}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {a (a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {\frac {\frac {a^2 (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {a (a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {(a-b) \sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\) |
-1/3*(Cos[e + f*x]*Sin[e + f*x])/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (-(((a - b)*Cos[e + f*x]*Sin[e + f*x])/((a + b)*f*Sqrt[a + b*Sin[e + f*x ]^2])) + (-((a*(a - b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^ 2])/(b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a^2*(a + b)*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]) )/(a*(a + b)))/(3*a*(a + b))
3.2.68.3.1 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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x ]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* a*(a + b)*(p + 1)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
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 = 2.74 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.19
| method | result | size |
| default | \(\frac {\left (a \,b^{2}-b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-2 a^{2} b -a \,b^{2}+b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a b \left (F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a +b \right )^{2} a b \cos \left (f x +e \right ) f}\) | \(483\) |
1/3*((a*b^2-b^3)*cos(f*x+e)^4*sin(f*x+e)+(-2*a^2*b-a*b^2+b^3)*cos(f*x+e)^2 *sin(f*x+e)-(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*a*b*(El lipticF(sin(f*x+e),(-1/a*b)^(1/2))*a+EllipticF(sin(f*x+e),(-1/a*b)^(1/2))* b-EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a+EllipticE(sin(f*x+e),(-1/a*b)^(1/ 2))*b)*cos(f*x+e)^2+(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2) *EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^3+2*(cos(f*x+e)^2)^(1/2)*(-b/a*cos (f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b+(cos(f *x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/ a*b)^(1/2))*a*b^2-(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*E llipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^3+(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x +e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2)/(a+b*sin( f*x+e)^2)^(3/2)/(a+b)^2/a/b/cos(f*x+e)/f
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 1400, normalized size of antiderivative = 6.33 \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/6*((2*((-I*a*b^3 + I*b^4)*cos(f*x + e)^4 - I*a^3*b - I*a^2*b^2 + I*a*b^3 + I*b^4 - 2*(-I*a^2*b^2 + I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b )/b^2) - ((2*I*a^2*b^2 - I*a*b^3 - I*b^4)*cos(f*x + e)^4 + 2*I*a^4 + 3*I*a ^3*b - I*a^2*b^2 - 3*I*a*b^3 - I*b^4 + 2*(-2*I*a^3*b - I*a^2*b^2 + 2*I*a*b ^3 + I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2* a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)* (cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*s qrt((a^2 + a*b)/b^2))/b^2) + (2*((I*a*b^3 - I*b^4)*cos(f*x + e)^4 + I*a^3* b + I*a^2*b^2 - I*a*b^3 - I*b^4 - 2*(I*a^2*b^2 - I*b^4)*cos(f*x + e)^2)*sq rt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^2*b^2 + I*a*b^3 + I*b^4)*cos(f*x + e)^4 - 2*I*a^4 - 3*I*a^3*b + I*a^2*b^2 + 3*I*a*b^3 + I*b^4 + 2*(2*I*a^3*b + I*a^2*b^2 - 2*I*a*b^3 - I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt ((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a* b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b ^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 4*(((I*a*b^3 + I*b^4)*c os(f*x + e)^4 + I*a^3*b + 3*I*a^2*b^2 + 3*I*a*b^3 + I*b^4 + 2*(-I*a^2*b^2 - 2*I*a*b^3 - I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) + ((-2 *I*a^2*b^2 - I*a*b^3)*cos(f*x + e)^4 - 2*I*a^4 - 5*I*a^3*b - 4*I*a^2*b^2 - I*a*b^3 + 2*(2*I*a^3*b + 3*I*a^2*b^2 + I*a*b^3)*cos(f*x + e)^2)*sqrt(-b)) *sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((...
Timed out. \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]