Integrand size = 21, antiderivative size = 27 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {-1+3 x+x^3}}\right )}{\sqrt {3}} \]
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {-1+3 x+x^3}}\right )}{\sqrt {3}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.73 (sec) , antiderivative size = 1340, normalized size of antiderivative = 49.63, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {7293, 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.
| \(\displaystyle \int \frac {x+2}{(x-1) \sqrt {x^3+3 x-1}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3}{(x-1) \sqrt {x^3+3 x-1}}+\frac {1}{\sqrt {x^3+3 x-1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 i 2^{5/6} \sqrt {\frac {x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {x^2-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {i \left (-2 x-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right ),\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {x^3+3 x-1}}-\frac {3 \sqrt [6]{\frac {2}{1+\sqrt {5}}} \sqrt {6-3 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}+i \sqrt [6]{2} \sqrt [3]{1+\sqrt {5}} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}} \sqrt {x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}} \sqrt {1-\frac {2 \left (x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}\right )}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1-\frac {2 \left (x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}\right )}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \operatorname {EllipticPi}\left (\frac {3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}{2 \left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )},\arcsin \left (\frac {2^{5/6} \sqrt [6]{1+\sqrt {5}} \sqrt {x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}}}{\sqrt {6-3 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}+i \sqrt [6]{2} \sqrt [3]{1+\sqrt {5}} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right ),\frac {3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}\right )}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) \sqrt {x^3+3 x-1}}\) |
((2*I)*2^(5/6)*Sqrt[((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x )/(6/(1 + Sqrt[5])^(1/3) - 3*(2*(1 + Sqrt[5]))^(1/3) - I*2^(1/6)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))])]*Sqrt[1 + (2 /(1 + Sqrt[5]))^(2/3) + ((1 + Sqrt[5])/2)^(2/3) - ((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3))*x + x^2]*EllipticF[ArcSin[Sqrt[(I*((2/(1 + Sqr t[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) - I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2 /3) + 3*((1 + Sqrt[5])/2)^(2/3)] - 2*x))/Sqrt[6*(4 + 2*(2/(1 + Sqrt[5]))^( 2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))]]], (2*2^(1/6)*Sqrt[3*(4 + 2*(2/(1 + S qrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))])/((6*I)/(1 + Sqrt[5])^(1/3) - (3*I)*(2*(1 + Sqrt[5]))^(1/3) + 2^(1/6)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5])) ^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))])])/Sqrt[-1 + 3*x + x^3] - (3*(2/(1 + Sqrt[5]))^(1/6)*Sqrt[6 - 3*2^(1/3)*(1 + Sqrt[5])^(2/3) + I*2^(1/6)*(1 + Sqrt[5])^(1/3)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5 ])^(2/3))]]*Sqrt[(2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x]*Sq rt[1 - (2*((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x))/(3*(2/( 1 + Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5])/2)^(1/3) - I*Sqrt[6 + 3*(2/(1 + Sqr t[5]))^(2/3) + 3*((1 + Sqrt[5])/2)^(2/3)])]*Sqrt[1 - (2*((2/(1 + Sqrt[5])) ^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x))/(3*(2/(1 + Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5])/2)^(1/3) + I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2/3) + 3*((1 + Sqrt[ 5])/2)^(2/3)])]*EllipticPi[(3*(2/(1 + Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5]...
3.3.99.3.1 Defintions of rubi rules used
Time = 2.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+3 x -1}\, \sqrt {3}}{3 x}\right )}{3}\) | \(25\) |
| pseudoelliptic | \(-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+3 x -1}\, \sqrt {3}}{3 x}\right )}{3}\) | \(25\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -6 \sqrt {x^{3}+3 x -1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (x -1\right )^{3}}\right )}{3}\) | \(69\) |
| elliptic | \(\text {Expression too large to display}\) | \(1075\) |
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{6} + 18 \, x^{5} + 15 \, x^{4} + 52 \, x^{3} - 4 \, \sqrt {3} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} - x\right )} \sqrt {x^{3} + 3 \, x - 1} - 9 \, x^{2} - 6 \, x + 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right ) \]
1/6*sqrt(3)*log((x^6 + 18*x^5 + 15*x^4 + 52*x^3 - 4*sqrt(3)*(x^4 + 3*x^3 + 3*x^2 - x)*sqrt(x^3 + 3*x - 1) - 9*x^2 - 6*x + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1))
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\int \frac {x + 2}{\left (x - 1\right ) \sqrt {x^{3} + 3 x - 1}}\, dx \]
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]
Time = 6.23 (sec) , antiderivative size = 1872, normalized size of antiderivative = 69.33 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\text {Too large to display} \]
(2*(-(x + 1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))/((3^(1/2)*( 1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2) /2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)*ellipticF(asin((( x + (3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)/((3^(1/2)*(1/( 5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)), -(3^(1/2)*((3^(1/2 )*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1 /2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2)*1i)/(3*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))))*((x + (3^(1/2)*(1/(5^(1/2)/2 + 1 /2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1 /2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2) ^(1/3))/2)*((3^(1/2)*(x - (3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^( 1/3)/2)*1i)/(3*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))))^(1/ 2))/(x^3 - x*((1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))*((3^(1 /2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2*...