Integrand size = 29, antiderivative size = 403 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\frac {2 (c-d) \sqrt {c+d} E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{(3-b) \sqrt {3+b} (b c-3 d) f}+\frac {2 \sqrt {3+b} (c-d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(3-b) (b c-3 d) \sqrt {c+d} f} \]
2*(c-d)*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin( f*x+e))^(1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e)) *(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((- a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/(a-b)/(-a*d+b*c)/f/( a+b)^(1/2)+2*(c-d)*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2 )/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d* sin(f*x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e))) ^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)/(a-b)/(-a *d+b*c)/f/(c+d)^(1/2)
Time = 8.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\frac {2 \sqrt {\frac {6-2 b}{3+b}} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {3-b}{3+b}} \cos \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{\sqrt {\frac {3+b \sin (e+f x)}{3+b}}}\right )|\frac {2 (b c-3 d)}{(-3+b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{(-3+b) f \sqrt {\frac {(3+b) (1+\sin (e+f x))}{3+b \sin (e+f x)}} \sqrt {3+b \sin (e+f x)} \sqrt {\frac {3+b \sin (e+f x)}{3+b}} \sqrt {\frac {(3+b) (c+d \sin (e+f x))}{(c+d) (3+b \sin (e+f x))}}} \]
(2*Sqrt[(6 - 2*b)/(3 + b)]*Cos[(2*e - Pi + 2*f*x)/4]*EllipticE[ArcSin[(Sqr t[(3 - b)/(3 + b)]*Cos[(2*e + Pi + 2*f*x)/4])/Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]], (2*(b*c - 3*d))/((-3 + b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/((- 3 + b)*f*Sqrt[((3 + b)*(1 + Sin[e + f*x]))/(3 + b*Sin[e + f*x])]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]*Sqrt[((3 + b)*(c + d*Si n[e + f*x]))/((c + d)*(3 + b*Sin[e + f*x]))])
Time = 0.91 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3274, 3042, 3297, 3475}
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.
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3274 |
| \(\displaystyle \frac {(c-d) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {(b c-a d) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c-d) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {(b c-a d) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {2 \sqrt {a+b} (c-d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (a-b) \sqrt {c+d} (b c-a d)}-\frac {(b c-a d) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {2 \sqrt {a+b} (c-d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (a-b) \sqrt {c+d} (b c-a d)}+\frac {2 (c-d) \sqrt {c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (a-b) \sqrt {a+b} (b c-a d)}\) |
(2*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f* x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*( c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b *Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)*f) + (2*Sqrt[a + b]*(c - d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b )*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d) *f)
3.8.91.3.1 Defintions of rubi rules used
Int[Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]/((a_.) + (b_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[(c - d)/(a - b) Int[1/(Sqrt[a + b*Si n[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Simp[(b*c - a*d)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Leaf count of result is larger than twice the leaf count of optimal. \(41769\) vs. \(2(379)=758\).
Time = 8.09 (sec) , antiderivative size = 41770, normalized size of antiderivative = 103.65
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\left (a + b \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]