Integrand size = 27, antiderivative size = 270 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{d (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^3 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 (c+d) f}-\frac {4 a^3 \left (4 c^2-5 c d-3 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^3 (4 c-5 d) (c-d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
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)*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)*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 = 6.00 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.87 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 a^3 (1+\sin (e+f x))^3 \left (-2 \left (4 c^3-c^2 d-8 c d^2-3 d^3\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+2 \left (4 c^3-5 c^2 d-4 c d^2+5 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d \cos (e+f x) \left (4 c^2-5 c d+3 d^2+d (c+d) \sin (e+f x)\right )\right )}{3 d^3 (c+d) f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]
Output:
(-2*a^3*(1 + Sin[e + f*x])^3*(-2*(4*c^3 - c^2*d - 8*c*d^2 - 3*d^3)*Ellipti cE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d) ] + 2*(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)] + d*Cos[e + f*x]*(4*c^2 - 5*c*d + 3*d^2 + d*(c + d)*Sin[e + f*x])))/(3*d^3*(c + d)*f*(Cos[(e + f* x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Sin[e + f*x]])
Time = 1.52 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.04, 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 |
\(\displaystyle \frac {2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (\frac {\frac {2 a^2 \left (4 c^2-5 c d-3 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 (4 c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{3 d}+\frac {2 a^2 (2 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\right )}{d (c+d)}\) |
Input:
Int[(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2),x]
Output:
(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))
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. \(2(257)=514\).
Time = 4.46 (sec) , antiderivative size = 1031, normalized size of antiderivative = 3.82
method | result | size |
default | \(\text {Expression too large to display}\) | \(1031\) |
parts | \(\text {Expression too large to display}\) | \(2574\) |
Input:
int((a+sin(f*x+e)*a)^3/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(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-16*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*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)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*Ell ipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^3+16*((c+d*si n(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+10*(-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+14*((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)*Elliptic E(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^2-10*(-d*(-1+s in(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*si n(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^3-6*((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)...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.80 \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")
Output:
2/9*(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 + (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(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^4 - 10*a^3*c^3*d - 9*a^3*c^2*d^2 + 15*a^3*c*d^3 + (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(-1/2*I*d)*weierstr assPInverse(-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^3*d + 5*I*a^3*c^2*d^2 + 3*I*a^3*c*d^3 + (-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(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^3*d - 5*I*a^3*c^2*d^2 - 3*I*a^3*c *d^3 + (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( -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*((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) + (c^2*d^4 + c*d^5)*f)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**(3/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(3/2), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(3/2), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(3/2),x)
Output:
int((a + a*sin(e + f*x))^3/(c + d*sin(e + f*x))^(3/2), x)
\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=a^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{2} d^{2}+2 \sin \left (f x +e \right ) c d +c^{2}}d x +\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{2} d^{2}+2 \sin \left (f x +e \right ) c d +c^{2}}d x +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{2} d^{2}+2 \sin \left (f x +e \right ) c d +c^{2}}d x \right )+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{2} d^{2}+2 \sin \left (f x +e \right ) c d +c^{2}}d x \right )\right ) \] Input:
int((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x)
Output:
a**3*(int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**2*d**2 + 2*sin(e + f*x)* c*d + c**2),x) + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3)/(sin(e + f *x)**2*d**2 + 2*sin(e + f*x)*c*d + c**2),x) + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)**2*d**2 + 2*sin(e + f*x)*c*d + c**2),x) + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)**2*d**2 + 2 *sin(e + f*x)*c*d + c**2),x))