Integrand size = 23, antiderivative size = 73 \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\frac {\left (5+(-1+x)^2\right ) (-1+x)}{24 \sqrt {3-2 (-1+x)^2-(-1+x)^4}}+\frac {E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )}{8 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{4 \sqrt {3}} \]
-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 = 22.67 (sec) , antiderivative size = 261, normalized size of antiderivative = 3.58 \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\frac {\sqrt {-x \left (-8+8 x-4 x^2+x^3\right )} \left (\frac {\sqrt {2} \left (-i+\sqrt {3}\right ) \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {\frac {4-2 x+x^2}{x^2}}}-\frac {2+x^2-4 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x^2 \sqrt {\frac {4-2 x+x^2}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{4-2 x+x^2}\right )}{24 (-2+x) x} \]
(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*((Sqrt[2]*(-I + Sqrt[3])*Sqrt[((-I)*( -2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/ (Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[(4 - 2*x + x^2)/x^2] - (2 + x^2 - (4*I)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*S qrt[(4 - 2*x + x^2)/x^2]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqr t[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(4 - 2*x + x^2)))/(24*(-2 + x )*x)
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, 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.
| \(\displaystyle \int \frac {1}{\left (-x^4+4 x^3-8 x^2+8 x\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 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
3.8.67.3.1 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. 931 vs. \(2 (61 ) = 122\).
Time = 2.03 (sec) , antiderivative size = 932, normalized size of antiderivative = 12.77
| method | result | size |
| risch | \(\frac {x^{3}-3 x^{2}+8 x -6}{24 \sqrt {-x \left (x^{3}-4 x^{2}+8 x -8\right )}}+\frac {\left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{6 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}-\frac {x \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )+2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (\frac {\left (6+2 i \sqrt {3}\right ) F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-2+2 i \sqrt {3}}+\frac {\left (-1+i \sqrt {3}\right ) E\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{2}-\frac {4 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \frac {-1-i \sqrt {3}}{1-i \sqrt {3}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{-1+i \sqrt {3}}\right )}{24 \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}+\frac {\left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (2 F\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )-2 \Pi \left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \frac {1+i \sqrt {3}}{-1+i \sqrt {3}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )\right )}{6 \left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(932\) |
| default | \(\text {Expression too large to display}\) | \(963\) |
| elliptic | \(\text {Expression too large to display}\) | \(963\) |
1/24*(x^3-3*x^2+8*x-6)/(-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))-1/24*(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))*EllipticE(((-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))*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))))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-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(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=-\frac {5 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) - 6 \, \sqrt {2} {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} {\rm weierstrassZeta}\left (-\frac {2}{3}, \frac {7}{54}, {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right )\right ) + 3 \, \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x} {\left (x^{2} + 2\right )}}{72 \, {\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )}} \]
-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)
\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (8 x-8 x^2+4 x^3-x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-x^4+4\,x^3-8\,x^2+8\,x\right )}^{3/2}} \,d x \]