Integrand size = 25, antiderivative size = 218 \[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {(3 a+4 b) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{15 f}-\frac {\cos (e+f x) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}+\frac {\left (3 a^2+13 a b+8 b^2\right ) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{15 b f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a (a+b) (3 a+4 b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{15 b f \sqrt {a+b \sin ^2(e+f x)}} \]
-1/5*cos(f*x+e)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2)/f-1/15*(3*a+4*b)*cos(f *x+e)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/f+1/15*(3*a^2+13*a*b+8*b^2)*(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)/b/f/(1+b*sin(f*x+e)^2/a)^(1/2)-1/15*a*(a+b)*(3*a+4*b)*(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/f/(a+b*sin(f*x+e)^2)^(1/2)
Time = 1.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.92 \[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {16 a \left (3 a^2+13 a b+8 b^2\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-16 a \left (3 a^2+7 a b+4 b^2\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (48 a^2+68 a b+25 b^2-4 b (9 a+7 b) \cos (2 (e+f x))+3 b^2 \cos (4 (e+f x))\right ) \sin (2 (e+f x))}{240 b f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(16*a*(3*a^2 + 13*a*b + 8*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*Elli pticE[e + f*x, -(b/a)] - 16*a*(3*a^2 + 7*a*b + 4*b^2)*Sqrt[(2*a + b - b*Co s[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(48*a^2 + 68*a*b + 25*b^2 - 4*b*(9*a + 7*b)*Cos[2*(e + f*x)] + 3*b^2*Cos[4*(e + f*x)])*Sin [2*(e + f*x)])/(240*b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])
Time = 1.17 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3649, 3042, 3649, 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.
| \(\displaystyle \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (e+f x)^2 \left (a+b \sin (e+f x)^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {1}{5} \int \sqrt {b \sin ^2(e+f x)+a} \left ((3 a+4 b) \sin ^2(e+f x)+a\right )dx-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \sqrt {b \sin (e+f x)^2+a} \left ((3 a+4 b) \sin (e+f x)^2+a\right )dx-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)+2 a (3 a+2 b)}{\sqrt {b \sin ^2(e+f x)+a}}dx-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {\left (3 a^2+13 b a+8 b^2\right ) \sin (e+f x)^2+2 a (3 a+2 b)}{\sqrt {b \sin (e+f x)^2+a}}dx-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \sqrt {b \sin ^2(e+f x)+a}dx}{b}-\frac {a (a+b) (3 a+4 b) \int \frac {1}{\sqrt {b \sin ^2(e+f x)+a}}dx}{b}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \sqrt {b \sin (e+f x)^2+a}dx}{b}-\frac {a (a+b) (3 a+4 b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3657 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}dx}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin (e+f x)^2}{a}+1}dx}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \int \frac {1}{\sqrt {b \sin (e+f x)^2+a}}dx}{b}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3662 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}dx}{b \sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin (e+f x)^2}{a}+1}}dx}{b \sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{b f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{b f \sqrt {a+b \sin ^2(e+f x)}}\right )-\frac {(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}\right )-\frac {\sin (e+f x) \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{5 f}\) |
-1/5*(Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/f + (-1/3*(( 3*a + 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f + (((3* a^2 + 13*a*b + 8*b^2)*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*(a + b)*(3*a + 4*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]))/3)/5
3.2.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b* Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Simp[1/(2*(p + 1)) Int[(a + b*Sin[ e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a* p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && G tQ[p, 0]
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 = 3.54 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(-\frac {-3 b^{3} \left (\sin ^{7}\left (f x +e \right )\right )-9 a \,b^{2} \left (\sin ^{5}\left (f x +e \right )\right )-b^{3} \left (\sin ^{5}\left (f x +e \right )\right )+3 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+7 a^{2} \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b +4 a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-3 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-13 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-6 a^{2} b \left (\sin ^{3}\left (f x +e \right )\right )+5 a \,b^{2} \left (\sin ^{3}\left (f x +e \right )\right )+4 b^{3} \left (\sin ^{3}\left (f x +e \right )\right )+6 a^{2} b \sin \left (f x +e \right )+4 a \,b^{2} \sin \left (f x +e \right )}{15 b \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(429\) |
-1/15*(-3*b^3*sin(f*x+e)^7-9*a*b^2*sin(f*x+e)^5-b^3*sin(f*x+e)^5+3*(cos(f* x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^( 1/2))*a^3+7*a^2*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*Elliptic F(sin(f*x+e),(-1/a*b)^(1/2))*b+4*a*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2 )/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-3*(cos(f*x+e)^2)^(1/2) *((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^3-13* (cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-1 /a*b)^(1/2))*a^2*b-8*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*Ell ipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2-6*a^2*b*sin(f*x+e)^3+5*a*b^2*sin(f *x+e)^3+4*b^3*sin(f*x+e)^3+6*a^2*b*sin(f*x+e)+4*a*b^2*sin(f*x+e))/b/cos(f* x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f
\[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{2} \,d x } \]
integral((b*cos(f*x + e)^4 - (a + 2*b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos (f*x + e)^2 + a + b), x)
Timed out. \[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{2} \,d x } \]
\[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \sin ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int {\sin \left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]