Integrand size = 29, antiderivative size = 80 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x-3 \sqrt {1+x+x^2}+\frac {5}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right )+4 \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )-\text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x) \] Output:
x+5/2*arcsinh(1/3*(1+2*x)*3^(1/2))+4*arctanh(1/2*(1-x)/(x^2+x+1)^(1/2))-ar ctanh(1/2*(2+x)/(x^2+x+1)^(1/2))+ln(x)-4*ln(1+x)-3*(x^2+x+1)^(1/2)
Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x-3 \sqrt {1+x+x^2}-8 \log \left (-2-x+\sqrt {1+x+x^2}\right )+2 \log \left (-1-x+\sqrt {1+x+x^2}\right )+\frac {1}{2} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \] Input:
Integrate[(-3*x + Sqrt[1 + x + x^2])/(-1 + Sqrt[1 + x + x^2]),x]
Output:
x - 3*Sqrt[1 + x + x^2] - 8*Log[-2 - x + Sqrt[1 + x + x^2]] + 2*Log[-1 - x + Sqrt[1 + x + x^2]] + Log[-1 - 2*x + 2*Sqrt[1 + x + x^2]]/2
Time = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int \frac {\sqrt {x^2+x+1}-3 x}{\sqrt {x^2+x+1}-1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^2+x+1}}{\sqrt {x^2+x+1}-1}-\frac {3 x}{\sqrt {x^2+x+1}-1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5}{2} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right )+4 \text {arctanh}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right )-\text {arctanh}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )-3 \sqrt {x^2+x+1}+x+\log (x)-4 \log (x+1)\) |
Input:
Int[(-3*x + Sqrt[1 + x + x^2])/(-1 + Sqrt[1 + x + x^2]),x]
Output:
x - 3*Sqrt[1 + x + x^2] + (5*ArcSinh[(1 + 2*x)/Sqrt[3]])/2 + 4*ArcTanh[(1 - x)/(2*Sqrt[1 + x + x^2])] - ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])] + Log [x] - 4*Log[1 + x]
Time = 0.38 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00
method | result | size |
default | \(\ln \left (x \right )-4 \ln \left (1+x \right )+x +\sqrt {x^{2}+x +1}+\frac {5 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}-\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )-4 \sqrt {\left (1+x \right )^{2}-x}+4 \,\operatorname {arctanh}\left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )\) | \(80\) |
trager | \(-1+x -3 \sqrt {x^{2}+x +1}-\frac {\ln \left (\frac {8+96 x -308624 x^{12}-8448 x^{14}-2392341 x^{10}-4224608 x^{9}-1008642 x^{11}-64992 x^{13}-5593140 x^{8}-5493060 x^{7}-3865870 x^{6}-512 x^{15}+8 \sqrt {x^{2}+x +1}-445596 x^{4}+507 x^{2}+1526 x^{3}-1790544 x^{5}+450424 x^{4} \sqrt {x^{2}+x +1}+1264 x^{3} \sqrt {x^{2}+x +1}+458 x^{2} \sqrt {x^{2}+x +1}+92 x \sqrt {x^{2}+x +1}+512 \sqrt {x^{2}+x +1}\, x^{14}+8192 \sqrt {x^{2}+x +1}\, x^{13}+60704 \sqrt {x^{2}+x +1}\, x^{12}+275296 \sqrt {x^{2}+x +1}\, x^{11}+849754 \sqrt {x^{2}+x +1}\, x^{10}+1875388 \sqrt {x^{2}+x +1}\, x^{9}+3018000 \sqrt {x^{2}+x +1}\, x^{8}+3530640 \sqrt {x^{2}+x +1}\, x^{7}+2914860 \sqrt {x^{2}+x +1}\, x^{6}+1571080 \sqrt {x^{2}+x +1}\, x^{5}}{x^{4}}\right )}{2}\) | \(286\) |
Input:
int((-3*x+(x^2+x+1)^(1/2))/(-1+(x^2+x+1)^(1/2)),x,method=_RETURNVERBOSE)
Output:
ln(x)-4*ln(1+x)+x+(x^2+x+1)^(1/2)+5/2*arcsinh(2/3*3^(1/2)*(x+1/2))-arctanh (1/2*(2+x)/(x^2+x+1)^(1/2))-4*((1+x)^2-x)^(1/2)+4*arctanh(1/2*(1-x)/((1+x) ^2-x)^(1/2))
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.24 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x - 3 \, \sqrt {x^{2} + x + 1} - 4 \, \log \left (x + 1\right ) + \log \left (x\right ) - \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) + 4 \, \log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) - 4 \, \log \left (-x + \sqrt {x^{2} + x + 1} - 2\right ) - \frac {5}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \] Input:
integrate((-3*x+(x^2+x+1)^(1/2))/(-1+(x^2+x+1)^(1/2)),x, algorithm="fricas ")
Output:
x - 3*sqrt(x^2 + x + 1) - 4*log(x + 1) + log(x) - log(-x + sqrt(x^2 + x + 1) + 1) + 4*log(-x + sqrt(x^2 + x + 1)) + log(-x + sqrt(x^2 + x + 1) - 1) - 4*log(-x + sqrt(x^2 + x + 1) - 2) - 5/2*log(-2*x + 2*sqrt(x^2 + x + 1) - 1)
\[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=- \int \frac {3 x}{\sqrt {x^{2} + x + 1} - 1}\, dx - \int \left (- \frac {\sqrt {x^{2} + x + 1}}{\sqrt {x^{2} + x + 1} - 1}\right )\, dx \] Input:
integrate((-3*x+(x**2+x+1)**(1/2))/(-1+(x**2+x+1)**(1/2)),x)
Output:
-Integral(3*x/(sqrt(x**2 + x + 1) - 1), x) - Integral(-sqrt(x**2 + x + 1)/ (sqrt(x**2 + x + 1) - 1), x)
\[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=\int { -\frac {3 \, x - \sqrt {x^{2} + x + 1}}{\sqrt {x^{2} + x + 1} - 1} \,d x } \] Input:
integrate((-3*x+(x^2+x+1)^(1/2))/(-1+(x^2+x+1)^(1/2)),x, algorithm="maxima ")
Output:
3/4*x^2 + 1/2*x + integrate(-1/2*(3*x^3 + 2*x^2 - x)/(x^2 + x - 2*sqrt(x^2 + x + 1) + 2), x)
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x - 3 \, \sqrt {x^{2} + x + 1} - \frac {5}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) - 4 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) - \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) + 4 \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) - 4 \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 2 \right |}\right ) \] Input:
integrate((-3*x+(x^2+x+1)^(1/2))/(-1+(x^2+x+1)^(1/2)),x, algorithm="giac")
Output:
x - 3*sqrt(x^2 + x + 1) - 5/2*log(-2*x + 2*sqrt(x^2 + x + 1) - 1) - 4*log( abs(x + 1)) + log(abs(x)) - log(abs(-x + sqrt(x^2 + x + 1) + 1)) + 4*log(a bs(-x + sqrt(x^2 + x + 1))) + log(abs(-x + sqrt(x^2 + x + 1) - 1)) - 4*log (abs(-x + sqrt(x^2 + x + 1) - 2))
Timed out. \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x-4\,\ln \left (x+1\right )+\ln \left (x\right )-\int \frac {\left (3\,x-1\right )\,\sqrt {x^2+x+1}}{x\,\left (x+1\right )} \,d x \] Input:
int(-(3*x - (x + x^2 + 1)^(1/2))/((x + x^2 + 1)^(1/2) - 1),x)
Output:
x - 4*log(x + 1) + log(x) - int(((3*x - 1)*(x + x^2 + 1)^(1/2))/(x*(x + 1) ), x)
Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.60 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=-3 \sqrt {x^{2}+x +1}-3 \,\mathrm {log}\left (\frac {12 \sqrt {x^{2}+x +1}\, x -6 \sqrt {x^{2}+x +1}+12 x^{2}+6}{2 \sqrt {x^{2}+x +1}\, \sqrt {3}+2 \sqrt {3}\, x +\sqrt {3}}\right )-5 \,\mathrm {log}\left (\frac {6 \sqrt {x^{2}+x +1}+6 x}{\sqrt {3}}\right )+5 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}+x +1}+2 x -2}{\sqrt {3}}\right )+\frac {5 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}+x +1}+2 x +1}{\sqrt {3}}\right )}{2}+x +\frac {1}{2} \] Input:
int((-3*x+(x^2+x+1)^(1/2))/(-1+(x^2+x+1)^(1/2)),x)
Output:
( - 6*sqrt(x**2 + x + 1) - 6*log((12*sqrt(x**2 + x + 1)*x - 6*sqrt(x**2 + x + 1) + 12*x**2 + 6)/(2*sqrt(x**2 + x + 1)*sqrt(3) + 2*sqrt(3)*x + sqrt(3 ))) - 10*log((6*sqrt(x**2 + x + 1) + 6*x)/sqrt(3)) + 10*log((2*sqrt(x**2 + x + 1) + 2*x - 2)/sqrt(3)) + 5*log((2*sqrt(x**2 + x + 1) + 2*x + 1)/sqrt( 3)) + 2*x + 1)/2