Integrand size = 26, antiderivative size = 19 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \sqrt {-1+x^3}}{1+x+x^2} \]
Time = 0.64 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \sqrt {-1+x^3}}{1+x+x^2} \]
Result contains higher order function than in optimal. Order 4 vs. order 2 in optimal.
Time = 2.38 (sec) , antiderivative size = 719, normalized size of antiderivative = 37.84, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {7279, 2009}
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.
\(\displaystyle \int \frac {x^2-2 x-2}{\left (x^2+x+1\right ) \sqrt {x^3-1}} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {1}{\sqrt {x^3-1}}-\frac {3 (x+1)}{\left (x^2+x+1\right ) \sqrt {x^3-1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt {3}+i\right ) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} (1-x) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \sqrt [4]{3} \left (-\sqrt {3}+i\right ) \sqrt {2-\sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} (1-x) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\left (3+(2-i) \sqrt {3}\right ) \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \sqrt {2-\sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} (1-x) \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {6 \sqrt [4]{3} \left (\sqrt {3}+(-2+i)\right ) \left (\sqrt {3}+i\right ) \sqrt {26+15 \sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (1-x) E\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\left (3+(2+i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {6 \sqrt [4]{3} \left (1+i \sqrt {3}\right ) \left (\sqrt {3}+(-2-i)\right ) \sqrt {26+15 \sqrt {3}} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (1-x) E\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\left (3 i+(1+2 i) \sqrt {3}\right )^3 \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {2 (1-x)}{\sqrt {x^3-1}}\) |
(2*(1 - x))/Sqrt[-1 + x^3] + (6*3^(1/4)*(1 + I*Sqrt[3])*((-2 - I) + Sqrt[3 ])*Sqrt[26 + 15*Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*E llipticE[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/((3 *I + (1 + 2*I)*Sqrt[3])^3*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3] ) - (6*3^(1/4)*((-2 + I) + Sqrt[3])*(I + Sqrt[3])*Sqrt[26 + 15*Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 - Sqrt[3 ] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/((3 + (2 + I)*Sqrt[3])^3*Sqrt[ (1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3]) - (2*Sqrt[2 - Sqrt[3]]*(1 - x )*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[ 3] - x)^2)]*Sqrt[-1 + x^3]) - (2*3^(1/4)*(I - Sqrt[3])*Sqrt[2 - Sqrt[3]]*( 1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[ 3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/((3 + (2 - I)*Sqrt[3])*Sqrt[- ((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) + (2*3^(1/4)*Sqrt[2 - Sqrt[ 3]]*(I + Sqrt[3])*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*Elliptic F[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/((3 + (2 + I)*Sqrt[3])*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])
3.2.59.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 3.67 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {2 \left (x -1\right )}{\sqrt {x^{3}-1}}\) | \(13\) |
risch | \(-\frac {2 \left (x -1\right )}{\sqrt {x^{3}-1}}\) | \(13\) |
default | \(-\frac {2 \left (x -1\right )}{\sqrt {\left (x -1\right ) \left (x^{2}+x +1\right )}}\) | \(18\) |
trager | \(-\frac {2 \sqrt {x^{3}-1}}{x^{2}+x +1}\) | \(18\) |
elliptic | \(-\frac {2 \left (x -1\right )}{\sqrt {\left (x -1\right ) \left (x^{2}+x +1\right )}}\) | \(18\) |
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \, \sqrt {x^{3} - 1}}{x^{2} + x + 1} \]
\[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int \frac {x^{2} - 2 x - 2}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + x + 1\right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \, \sqrt {x - 1}}{\sqrt {x^{2} + x + 1}} \]
\[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\int { \frac {x^{2} - 2 \, x - 2}{\sqrt {x^{3} - 1} {\left (x^{2} + x + 1\right )}} \,d x } \]
Time = 4.98 (sec) , antiderivative size = 276, normalized size of antiderivative = 14.53 \[ \int \frac {-2-2 x+x^2}{\left (1+x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (6+9\,\sin \left (2\,\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\right )\,\sqrt {\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}+1}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}-6\,x+\sqrt {3}\,x\,2{}\mathrm {i}+\sqrt {3}\,2{}\mathrm {i}-\sqrt {3}\,x^2\,4{}\mathrm {i}-\sqrt {3}\,\sin \left (2\,\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\right )\,\sqrt {\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}+1}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,3{}\mathrm {i}\right )}{6\,\sqrt {1-\frac {x-1}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}+1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
((-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2) *1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(3^(1/2)*x*2i - 6*x + 3^(1/2)* 2i - 3^(1/2)*x^2*4i + 9*sin(2*asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2) ))*((x - 1)/((3^(1/2)*1i)/2 + 3/2) + 1)^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2) - 3^(1/2)*sin(2*asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2))) *((x - 1)/((3^(1/2)*1i)/2 + 3/2) + 1)^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/ 2))^(1/2)*3i + 6))/(6*(1 - (x - 1)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x - 1)/ ((3^(1/2)*1i)/2 + 3/2) + 1)^(1/2)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2 ))