Integrand size = 32, antiderivative size = 15 \[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {1+x+x^3}}\right ) \] Output:
2*arctanh(x/(x^3+x+1)^(1/2))
Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {1+x+x^3}}\right ) \] Input:
Integrate[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]
Output:
2*ArcTanh[x/Sqrt[1 + x + x^3]]
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.
| \(\displaystyle \int \frac {-x^3+x+2}{\sqrt {x^3+x+1} \left (x^3-x^2+x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-x^2+2 x+3}{\sqrt {x^3+x+1} \left (x^3-x^2+x+1\right )}-\frac {1}{\sqrt {x^3+x+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {1}{\sqrt {x^3+x+1} \left (x^3-x^2+x+1\right )}dx+2 \int \frac {x}{\sqrt {x^3+x+1} \left (x^3-x^2+x+1\right )}dx-\int \frac {x^2}{\sqrt {x^3+x+1} \left (x^3-x^2+x+1\right )}dx-\frac {2 i \sqrt {\frac {6^{2/3} x-\sqrt [3]{2 \left (\sqrt {93}-9\right )}+2 \sqrt [3]{\frac {3}{\sqrt {93}-9}}}{6 \sqrt [3]{\frac {3}{\sqrt {93}-9}}-3 \sqrt [3]{2 \left (\sqrt {93}-9\right )}-i \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{\sqrt {93}-9}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (\sqrt {93}-9\right )\right )^{2/3}}}} \sqrt {18 x^2-6 \sqrt [3]{3} \left (\sqrt [3]{\frac {6}{\sqrt {93}-9}}-\sqrt [3]{\frac {1}{2} \left (\sqrt {93}-9\right )}\right ) x+\sqrt [3]{2} \left (3 \left (\sqrt {93}-9\right )\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{\sqrt {93}-9}\right )^{2/3}+6} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i \left (\sqrt [3]{6} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}-i \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}\right )-12 x\right )}}{2^{3/4} \sqrt [4]{3 \left (12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}\right ),\frac {2 \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}{i \left (6 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-3 \sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )+\sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}\right )}{3 \sqrt {x^3+x+1}}\) |
Input:
Int[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]
Output:
$Aborted
Time = 0.97 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{3}+x +1}}{x}\right )\) | \(16\) |
| pseudoelliptic | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{3}+x +1}}{x}\right )\) | \(16\) |
| trager | \(-\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+x +1}\, x -x^{2}-x -1}{x^{3}-x^{2}+x +1}\right )\) | \(45\) |
| elliptic | \(\text {Expression too large to display}\) | \(1905\) |
Input:
int((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x,method=_RETURNVERBOSE)
Output:
2*arctanh((x^3+x+1)^(1/2)/x)
Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\log \left (\frac {x^{3} + x^{2} + 2 \, \sqrt {x^{3} + x + 1} x + x + 1}{x^{3} - x^{2} + x + 1}\right ) \] Input:
integrate((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x, algorithm="fricas")
Output:
log((x^3 + x^2 + 2*sqrt(x^3 + x + 1)*x + x + 1)/(x^3 - x^2 + x + 1))
\[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=- \int \left (- \frac {x}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\right )\, dx - \int \frac {x^{3}}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\, dx - \int \left (- \frac {2}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\right )\, dx \] Input:
integrate((-x**3+x+2)/(x**3+x+1)**(1/2)/(x**3-x**2+x+1),x)
Output:
-Integral(-x/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x **3 + x + 1) + sqrt(x**3 + x + 1)), x) - Integral(x**3/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 + x + 1 )), x) - Integral(-2/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 + x + 1)), x)
\[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\int { -\frac {x^{3} - x - 2}{{\left (x^{3} - x^{2} + x + 1\right )} \sqrt {x^{3} + x + 1}} \,d x } \] Input:
integrate((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x, algorithm="maxima")
Output:
-integrate((x^3 - x - 2)/((x^3 - x^2 + x + 1)*sqrt(x^3 + x + 1)), x)
\[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\int { -\frac {x^{3} - x - 2}{{\left (x^{3} - x^{2} + x + 1\right )} \sqrt {x^{3} + x + 1}} \,d x } \] Input:
integrate((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x, algorithm="giac")
Output:
integrate(-(x^3 - x - 2)/((x^3 - x^2 + x + 1)*sqrt(x^3 + x + 1)), x)
Time = 8.53 (sec) , antiderivative size = 2490, normalized size of antiderivative = 166.00 \[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=\text {Too large to display} \] Input:
int((x - x^3 + 2)/((x + x^3 + 1)^(1/2)*(x - x^2 + x^3 + 1)),x)
Output:
symsum(-(2*(-(x + 1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2 )*108^(1/2))/108 - 1/2)^(1/3))/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^( 1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1 /3))/2))^(1/2)*((x + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3) ) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1 /2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)/((3^(1/ 2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/1 08 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3 *((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(3*((31^(1 /2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) *1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^ (1/2))/108 - 1/2)^(1/3))/2)*ellipticPi(((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2 ))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/( 2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)/(root(z^3 - z^2 + z + 1, z, k) + (3^(1/2)*(1/(3*((31^(1/2) *108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i )/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2)) /108 - 1/2)^(1/3)/2), asin(((x + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((...
\[ \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx=2 \left (\int \frac {\sqrt {x^{3}+x +1}}{x^{6}-x^{5}+2 x^{4}+x^{3}+2 x +1}d x \right )-\left (\int \frac {\sqrt {x^{3}+x +1}\, x^{3}}{x^{6}-x^{5}+2 x^{4}+x^{3}+2 x +1}d x \right )+\int \frac {\sqrt {x^{3}+x +1}\, x}{x^{6}-x^{5}+2 x^{4}+x^{3}+2 x +1}d x \] Input:
int((-x^3+x+2)/(x^3+x+1)^(1/2)/(x^3-x^2+x+1),x)
Output:
2*int(sqrt(x**3 + x + 1)/(x**6 - x**5 + 2*x**4 + x**3 + 2*x + 1),x) - int( (sqrt(x**3 + x + 1)*x**3)/(x**6 - x**5 + 2*x**4 + x**3 + 2*x + 1),x) + int ((sqrt(x**3 + x + 1)*x)/(x**6 - x**5 + 2*x**4 + x**3 + 2*x + 1),x)