Integrand size = 27, antiderivative size = 258 \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {8 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}+\frac {4 a^3 \left (4 c^2-15 c d+27 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
8/15*a^3*(c-3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-2/5*cos(f*x+e)*(a ^3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^(1/2)/d/f-4/15*a^3*(4*c^2-15*c*d+27*d^ 2)*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/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-4/15*a^3*(c-d)*(4*c^2-11 *c*d+15*d^2)*InverseJacobiAM(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 = 6.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.95 \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {a^3 (1+\sin (e+f x))^3 \left (4 \left (4 c^3-11 c^2 d+12 c d^2+27 d^3\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-4 \left (4 c^3-15 c^2 d+26 c d^2-15 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d \cos (e+f x) \left (8 c^2-30 c d-3 d^2+3 d^2 \cos (2 (e+f x))+2 (c-15 d) d \sin (e+f x)\right )\right )}{15 d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]
Output:
-1/15*(a^3*(1 + Sin[e + f*x])^3*(4*(4*c^3 - 11*c^2*d + 12*c*d^2 + 27*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 - 15*c^2*d + 26*c*d^2 - 15*d^3)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*Cos[e + f *x]*(8*c^2 - 30*c*d - 3*d^2 + 3*d^2*Cos[2*(e + f*x)] + 2*(c - 15*d)*d*Sin[ e + f*x])))/(d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[ e + f*x]])
Time = 1.45 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3242, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{\sqrt {c+d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{\sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int \frac {(\sin (e+f x) a+a) \left (a^2 (c+3 d)-2 a^2 (c-3 d) \sin (e+f x)\right )}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {(\sin (e+f x) a+a) \left (a^2 (c+3 d)-2 a^2 (c-3 d) \sin (e+f x)\right )}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {2 \int \frac {-2 (c-3 d) \sin ^2(e+f x) a^3+(c+3 d) a^3+\left (a^3 (c+3 d)-2 a^3 (c-3 d)\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {-2 (c-3 d) \sin (e+f x)^2 a^3+(c+3 d) a^3+\left (a^3 (c+3 d)-2 a^3 (c-3 d)\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \left (\frac {2 \int \frac {d (c+15 d) a^3+\left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {d (c+15 d) a^3+\left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {d (c+15 d) a^3+\left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 \left (\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \left (\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \left (\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \left (\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {4 a^3 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{5 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt {c+d \sin (e+f x)}}{5 d f}\) |
Input:
Int[(a + a*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]
Output:
(-2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(5*d*f ) + (2*((4*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) + ((2*a^3*(4*c^2 - 15*c*d + 27*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^3*(c - d)*(4*c^2 - 11*c*d + 15*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)))/(5*d)
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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, 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. \(1034\) vs. \(2(243)=486\).
Time = 3.05 (sec) , antiderivative size = 1035, normalized size of antiderivative = 4.01
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1035\) |
| parts | \(\text {Expression too large to display}\) | \(1152\) |
Input:
int((a+sin(f*x+e)*a)^3/(c+d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/15*a^3*(8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2 )*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2) ,((c-d)/(c+d))^(1/2))*c^3*d-36*c^2*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+ sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*s in(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^2+112*c*((c+d*sin(f*x+e))/( c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/ 2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^3-84*(( c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f *x+e))/(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)*(-d*(-1+sin(f*x+e))/(c+d))^(1 /2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/ 2),((c-d)/(c+d))^(1/2))*c^4+30*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin (f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d ))^(1/2))*((c+d*sin(f*x+e))/(c-d))^(1/2)*c^3*d-46*((c+d*sin(f*x+e))/(c-d)) ^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*El lipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^2-30*(-d *(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE((( c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c-d)) ^(1/2)*c*d^3+54*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^ (1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d)...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.07 \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2/45*(2*(8*a^3*c^3 - 30*a^3*c^2*d + 51*a^3*c*d^2 - 45*a^3*d^3)*sqrt(1/2*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) + 2*(8*a^3 *c^3 - 30*a^3*c^2*d + 51*a^3*c*d^2 - 45*a^3*d^3)*sqrt(-1/2*I*d)*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) + 6*(4*I*a^3*c^2*d - 15 *I*a^3*c*d^2 + 27*I*a^3*d^3)*sqrt(1/2*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)) + 6*(-4*I*a^3*c^2*d + 15*I*a^3*c*d^2 - 27*I *a^3*d^3)*sqrt(-1/2*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*(3*a^3*d^3*cos(f*x + e)*sin(f*x + e) - (4*a^3*c*d^2 - 1 5*a^3*d^3)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4*f)
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=a^{3} \left (\int \frac {3 \sin {\left (e + f x \right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx + \int \frac {3 \sin ^{2}{\left (e + f x \right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx + \int \frac {\sin ^{3}{\left (e + f x \right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx + \int \frac {1}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**(1/2),x)
Output:
a**3*(Integral(3*sin(e + f*x)/sqrt(c + d*sin(e + f*x)), x) + Integral(3*si n(e + f*x)**2/sqrt(c + d*sin(e + f*x)), x) + Integral(sin(e + f*x)**3/sqrt (c + d*sin(e + f*x)), x) + Integral(1/sqrt(c + d*sin(e + f*x)), x))
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3/sqrt(d*sin(f*x + e) + c), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^3/sqrt(d*sin(f*x + e) + c), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(1/2),x)
Output:
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(1/2), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx=a^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right ) d +c}d x +\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right ) d +c}d x +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right ) d +c}d x \right )+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right ) d +c}d x \right )\right ) \] Input:
int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x)
Output:
a**3*(int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)*d + c),x) + int((sqrt(sin (e + f*x)*d + c)*sin(e + f*x)**3)/(sin(e + f*x)*d + c),x) + 3*int((sqrt(si n(e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)*d + c),x) + 3*int((sqrt(s in(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)*d + c),x))