Integrand size = 27, antiderivative size = 251 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (c-d) \cos (e+f x) (27+27 \sin (e+f x))}{d (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {36 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 (c+d) f}-\frac {36 \left (4 c^2-5 c d-3 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d^3 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {36 (4 c-5 d) (c-d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d^3 f \sqrt {c+d \sin (e+f x)}} \]
2*(c-d)*cos(f*x+e)*(a^3+a^3*sin(f*x+e))/d/(c+d)/f/(c+d*sin(f*x+e))^(1/2)-4 /3*a^3*(2*c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/(c+d)/f+4/3*a^3*(4*c^ 2-5*c*d-3*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f* x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f *x+e))^(1/2)/d^3/(c+d)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-4/3*a^3*(4*c-5*d)* (c-d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellipt icF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/( c+d))^(1/2)/d^3/f/(c+d*sin(f*x+e))^(1/2)
Time = 0.73 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.78 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {9 \left (4 \left (4 c^3-c^2 d-8 c d^2-3 d^3\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-4 \left (4 c^3-5 c^2 d-4 c d^2+5 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-2 d \cos (e+f x) \left (4 c^2-5 c d+3 d^2+d (c+d) \sin (e+f x)\right )\right )}{d^3 (c+d) f \sqrt {c+d \sin (e+f x)}} \]
(9*(4*(4*c^3 - c^2*d - 8*c*d^2 - 3*d^3)*EllipticE[(-2*e + Pi - 2*f*x)/4, ( 2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 4*(4*c^3 - 5*c^2*d - 4* c*d^2 + 5*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d *Sin[e + f*x])/(c + d)] - 2*d*Cos[e + f*x]*(4*c^2 - 5*c*d + 3*d^2 + d*(c + d)*Sin[e + f*x])))/(d^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])
Time = 1.43 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3241, 3042, 3447, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c+d \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c+d \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3241 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \int \frac {(\sin (e+f x) a+a) (a (c-2 d)-a (2 c-d) \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}dx}{d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \int \frac {(\sin (e+f x) a+a) (a (c-2 d)-a (2 c-d) \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}dx}{d (c+d)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \int \frac {-\left ((2 c-d) \sin ^2(e+f x) a^2\right )+(c-2 d) a^2+\left (a^2 (c-2 d)-a^2 (2 c-d)\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \int \frac {-\left ((2 c-d) \sin (e+f x)^2 a^2\right )+(c-2 d) a^2+\left (a^2 (c-2 d)-a^2 (2 c-d)\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{d (c+d)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {2 \int \frac {(c-5 d) d a^2+\left (4 c^2-5 d c-3 d^2\right ) \sin (e+f x) a^2}{2 \sqrt {c+d \sin (e+f x)}}dx}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\int \frac {(c-5 d) d a^2+\left (4 c^2-5 d c-3 d^2\right ) \sin (e+f x) a^2}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\int \frac {(c-5 d) d a^2+\left (4 c^2-5 d c-3 d^2\right ) \sin (e+f x) a^2}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {2 a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {2 a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {2 a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {2 a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
(2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(d*(c + d)*f*Sqrt[c + d* Sin[e + f*x]]) - (2*a*((2*a^2*(2*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f* x]])/(3*d*f) + ((2*a^2*(4*c^2 - 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/ 2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x]) /(c + d)]) - (2*a^2*(4*c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(3*d)))/(d*(c + d))
3.5.100.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b *Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1030\) vs. \(2(318)=636\).
Time = 5.80 (sec) , antiderivative size = 1031, normalized size of antiderivative = 4.11
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1031\) |
| parts | \(\text {Expression too large to display}\) | \(2575\) |
-2/3*(8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d *(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c- d)/(c+d))^(1/2))*c^3*d-16*c^2*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e) -1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x +e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^2-8*c*((c+d*sin(f*x+e))/(c-d))^(1 /2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*Ellipt icF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^3+16*((c+d*sin(f *x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c- d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^ 4-8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(si n(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/( c+d))^(1/2))*c^4+10*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d ))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d) )^(1/2),((c-d)/(c+d))^(1/2))*c^3*d+14*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(si n(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d *sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^2-10*((c+d*sin(f*x+e) )/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^( 1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c*d^3-6 *((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f *x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 786, normalized size of antiderivative = 3.13 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} {\left (8 \, a^{3} c^{3} d - 10 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} + 15 \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (8 \, a^{3} c^{4} - 10 \, a^{3} c^{3} d - 9 \, a^{3} c^{2} d^{2} + 15 \, a^{3} c d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (8 \, a^{3} c^{3} d - 10 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} + 15 \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (8 \, a^{3} c^{4} - 10 \, a^{3} c^{3} d - 9 \, a^{3} c^{2} d^{2} + 15 \, a^{3} c d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 3 \, {\left (\sqrt {2} {\left (-4 i \, a^{3} c^{2} d^{2} + 5 i \, a^{3} c d^{3} + 3 i \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-4 i \, a^{3} c^{3} d + 5 i \, a^{3} c^{2} d^{2} + 3 i \, a^{3} c d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left (\sqrt {2} {\left (4 i \, a^{3} c^{2} d^{2} - 5 i \, a^{3} c d^{3} - 3 i \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (4 i \, a^{3} c^{3} d - 5 i \, a^{3} c^{2} d^{2} - 3 i \, a^{3} c d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left ({\left (a^{3} c d^{3} + a^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} + 3 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{9 \, {\left ({\left (c d^{5} + d^{6}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{4} + c d^{5}\right )} f\right )}} \]
2/9*((sqrt(2)*(8*a^3*c^3*d - 10*a^3*c^2*d^2 - 9*a^3*c*d^3 + 15*a^3*d^4)*si n(f*x + e) + sqrt(2)*(8*a^3*c^4 - 10*a^3*c^3*d - 9*a^3*c^2*d^2 + 15*a^3*c* d^3))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c ^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d ) + (sqrt(2)*(8*a^3*c^3*d - 10*a^3*c^2*d^2 - 9*a^3*c*d^3 + 15*a^3*d^4)*sin (f*x + e) + sqrt(2)*(8*a^3*c^4 - 10*a^3*c^3*d - 9*a^3*c^2*d^2 + 15*a^3*c*d ^3))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I* c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/ d) - 3*(sqrt(2)*(-4*I*a^3*c^2*d^2 + 5*I*a^3*c*d^3 + 3*I*a^3*d^4)*sin(f*x + e) + sqrt(2)*(-4*I*a^3*c^3*d + 5*I*a^3*c^2*d^2 + 3*I*a^3*c*d^3))*sqrt(I*d )*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^ 3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^ 2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) - 3*(sqrt( 2)*(4*I*a^3*c^2*d^2 - 5*I*a^3*c*d^3 - 3*I*a^3*d^4)*sin(f*x + e) + sqrt(2)* (4*I*a^3*c^3*d - 5*I*a^3*c^2*d^2 - 3*I*a^3*c*d^3))*sqrt(-I*d)*weierstrassZ eta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstras sPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3* (3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 3*((a^3*c*d^3 + a^3* d^4)*cos(f*x + e)*sin(f*x + e) + (4*a^3*c^2*d^2 - 5*a^3*c*d^3 + 3*a^3*d^4) *cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/((c*d^5 + d^6)*f*sin(f*x + e) ...
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]