Integrand size = 29, antiderivative size = 499 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {4 b (c-d) \sqrt {c+d} \left (6 b c-27 d+b^2 d\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{3 (3-b)^2 (3+b)^{3/2} (b c-3 d)^3 f}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 \left (9-b^2\right ) (b c-3 d) f (3+b \sin (e+f x))^{3/2}}+\frac {2 \left (9 b (c-d)-27 d+b^2 (c+2 d)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (3-b)^2 \sqrt {3+b} (b c-3 d)^2 \sqrt {c+d} f} \]
4/3*b*(c-d)*(-3*a^2*d+2*a*b*c+b^2*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)^2/(a+b)^(3/2)/(-a*d+b*c)^3/f+2/3*(3*a*b*(c-d)-3*a^2*d+b^2*( c+2*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*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)^2/(-a*d+b*c)^2/f/(a+b)^(1/ 2)/(c+d)^(1/2)+2/3*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/(-a*d+b *c)/f/(a+b*sin(f*x+e))^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(2040\) vs. \(2(499)=998\).
Time = 6.42 (sec) , antiderivative size = 2040, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\text {Result too large to show} \]
(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*b^2*Cos[e + f*x])/ (3*(-9 + b^2)*(b*c - 3*d)*(3 + b*Sin[e + f*x])^2) + (4*(6*b^3*c*Cos[e + f* x] - 27*b^2*d*Cos[e + f*x] + b^4*d*Cos[e + f*x]))/(3*(-9 + b^2)^2*(b*c - 3 *d)^2*(3 + b*Sin[e + f*x]))))/f + ((-4*(-(b*c) + 3*d)*(27*b^2*c^2 + b^4*c^ 2 - 162*b*c*d + 6*b^3*c*d + 243*d^2 - 45*b^2*d^2 + 2*b^4*d^2)*Sqrt[((c + d )*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Cs c[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], ( 2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2 ]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/( -(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(12*b^3*c^2 - 18*b^2*c*d + 2*b^4*c*d - 162*b* d^2 + 6*b^3*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*Ell ipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x ]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^ 2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f* x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b *Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqr...
Time = 1.45 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3042, 3281, 27, 3042, 3477, 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.
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}-\frac {2 \int -\frac {-3 d a^2+3 b c a+2 b^2 d-b (b c-3 a d) \sin (e+f x)}{2 (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-3 d a^2+3 b c a+2 b^2 d-b (b c-3 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-3 d a^2+3 b c a+2 b^2 d-b (b c-3 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {\left (-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {2 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{a-b}-\frac {2 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {2 \sqrt {a+b} \left (-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (a-b) \sqrt {c+d} (b c-a d)}-\frac {2 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{a-b}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {2 \sqrt {a+b} \left (-3 a^2 d+3 a b (c-d)+b^2 (c+2 d)\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (a-b) \sqrt {c+d} (b c-a d)}+\frac {4 b (c-d) \sqrt {c+d} \left (-3 a^2 d+2 a b c+b^2 d\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (a-b) \sqrt {a+b} (b c-a d)^2}}{3 \left (a^2-b^2\right ) (b c-a d)}+\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^{3/2}}\) |
(2*b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)*f *(a + b*Sin[e + f*x])^(3/2)) + ((4*b*(c - d)*Sqrt[c + d]*(2*a*b*c - 3*a^2* d + b^2*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)^2*f) + (2*Sqrt [a + b]*(3*a*b*(c - d) - 3*a^2*d + b^2*(c + 2*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 - Si n[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*(a^2 - b^2)*(b*c - a*d))
3.8.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 ))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2* n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 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)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{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] && NeQ[A, B]
Leaf count of result is larger than twice the leaf count of optimal. \(201689\) vs. \(2(476)=952\).
Time = 12.68 (sec) , antiderivative size = 201690, normalized size of antiderivative = 404.19
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^3*d*cos(f*x + e)^4 - (3*a*b^2*c + (3*a^2*b + 2*b^3)*d)*cos(f*x + e)^2 + (a^3 + 3*a*b^2 )*c + (3*a^2*b + b^3)*d - ((b^3*c + 3*a*b^2*d)*cos(f*x + e)^2 - (3*a^2*b + b^3)*c - (a^3 + 3*a*b^2)*d)*sin(f*x + e)), x)
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\left (a + b \sin {\left (e + f x \right )}\right )^{\frac {5}{2}} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
\[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]