Integrand size = 19, antiderivative size = 73 \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=\frac {\left (5+(-1+x)^2\right ) (-1+x)}{24 \sqrt {3-2 (-1+x)^2-(-1+x)^4}}+\frac {E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )}{8 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{4 \sqrt {3}} \]
-1/24*EllipticE(-1+x,1/3*I*3^(1/2))*3^(1/2)+1/12*EllipticF(-1+x,1/3*I*3^(1 /2))*3^(1/2)+1/24*(5+(-1+x)^2)*(-1+x)/(3-2*(-1+x)^2-(-1+x)^4)^(1/2)
Result contains complex when optimal does not.
Time = 19.72 (sec) , antiderivative size = 298, normalized size of antiderivative = 4.08 \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=\frac {(-2+x)^2 x \left (4-2 x+x^2\right ) \left (2 (-1+x) x-3 \left (4-2 x+x^2\right )-\frac {3 x \left (4-2 x+x^2\right )}{-2+x}-4 (2-x) \sqrt {\frac {4-2 x+x^2}{(-2+x)^2}} \left (x \sqrt {\frac {4-2 x+x^2}{(-2+x)^2}}-\sqrt {2} \left (i+\sqrt {3}\right ) \sqrt {\frac {i x}{\left (i+\sqrt {3}\right ) (-2+x)}} E\left (\arcsin \left (\frac {\sqrt {-i+\sqrt {3}-\frac {4 i}{-2+x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{i+\sqrt {3}}\right )+4 i \sqrt {2} \sqrt {\frac {i x}{\left (i+\sqrt {3}\right ) (-2+x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i+\sqrt {3}-\frac {4 i}{-2+x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{i+\sqrt {3}}\right )\right )\right )}{96 \left (-x \left (-8+8 x-4 x^2+x^3\right )\right )^{3/2}} \]
((-2 + x)^2*x*(4 - 2*x + x^2)*(2*(-1 + x)*x - 3*(4 - 2*x + x^2) - (3*x*(4 - 2*x + x^2))/(-2 + x) - 4*(2 - x)*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*(x*Sqr t[(4 - 2*x + x^2)/(-2 + x)^2] - Sqrt[2]*(I + Sqrt[3])*Sqrt[(I*x)/((I + Sqr t[3])*(-2 + x))]*EllipticE[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqr t[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])] + (4*I)*Sqrt[2]*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)] /(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])])))/(96*(-(x*(-8 + 8*x - 4* x^2 + x^3)))^(3/2))
Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2458, 1405, 27, 1494, 27, 399, 321, 327}
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.
\(\displaystyle \int \frac {1}{\left ((2-x) x \left (x^2-2 x+4\right )\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}d(x-1)\) |
\(\Big \downarrow \) 1405 |
\(\displaystyle \frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}-\frac {1}{48} \int -\frac {2 \left (3-(x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{24} \int \frac {3-(x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\) |
\(\Big \downarrow \) 1494 |
\(\displaystyle \frac {1}{12} \int \frac {3-(x-1)^2}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{24} \int \frac {3-(x-1)^2}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {1}{24} \left (6 \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)-\int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {1}{24} \left (-\int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)-2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {1}{24} \left (\sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt {-(x-1)^4-2 (x-1)^2+3}}\) |
((5 + (-1 + x)^2)*(-1 + x))/(24*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + (Sq rt[3]*EllipticE[ArcSin[1 - x], -1/3] - 2*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3])/24
3.8.72.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[(d + e*x^2)/(Sqr t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e }, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (61 ) = 122\).
Time = 1.45 (sec) , antiderivative size = 963, normalized size of antiderivative = 13.19
method | result | size |
default | \(\text {Expression too large to display}\) | \(963\) |
elliptic | \(\text {Expression too large to display}\) | \(963\) |
-1/32*(-x^3+4*x^2-8*x+8)/(x*(-x^3+4*x^2-8*x+8))^(1/2)+2*x*(1/24+1/192*x^2) /(-x*(x^3-4*x^2+8*x-8))^(1/2)+1/6*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^ (1/2))/(x-2))^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*(( x-1-I*3^(1/2))/(1+I*3^(1/2))/(x-2))^(1/2)/(-1+I*3^(1/2))/(-x*(x-2)*(x-1+I* 3^(1/2))*(x-1-I*3^(1/2)))^(1/2)*EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/ (x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^( 1/2))+1/6*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2)*(x-2 )^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1 /2))/(x-2))^(1/2)/(-1+I*3^(1/2))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2)) )^(1/2)*(2*EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^ (1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))-2*EllipticPi((( -1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),(1+I*3^(1/2))/(-1+I*3^(1/2)),(( 1+I*3^(1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2)))-1/24*(x* (x-1+I*3^(1/2))*(x-1-I*3^(1/2))+2*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^ (1/2))/(x-2))^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*(( x-1-I*3^(1/2))/(1+I*3^(1/2))/(x-2))^(1/2)*(1/2*(6+2*I*3^(1/2))/(-1+I*3^(1/ 2))*EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))* (-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))+1/2*(-1+I*3^(1/2))*Ell ipticE(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3 ^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))-4/(-1+I*3^(1/2))*EllipticP...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=-\frac {5 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) - 6 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} {\rm weierstrassZeta}\left (-\frac {2}{3}, \frac {7}{54}, {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right )\right ) + 3 \, \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x} {\left (x^{2} + 2\right )}}{72 \, {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )}} \]
-1/72*(5*sqrt(2)*(x^4 - 4*x^3 + 8*x^2 - 8*x)*weierstrassPInverse(-2/3, 7/5 4, -1/3*(x - 3)/x) - 6*sqrt(2)*(x^4 - 4*x^3 + 8*x^2 - 8*x)*weierstrassZeta (-2/3, 7/54, weierstrassPInverse(-2/3, 7/54, -1/3*(x - 3)/x)) + 3*sqrt(-x^ 4 + 4*x^3 - 8*x^2 + 8*x)*(x^2 + 2))/(x^4 - 4*x^3 + 8*x^2 - 8*x)
\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (2 - x\right ) \left (x^{2} - 2 x + 4\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=\int { \frac {1}{\left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=\int { \frac {1}{\left (-{\left (x^{2} - 2 \, x + 4\right )} {\left (x - 2\right )} x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx=\int \frac {1}{{\left (-x\,\left (x-2\right )\,\left (x^2-2\,x+4\right )\right )}^{3/2}} \,d x \]