Integrand size = 24, antiderivative size = 87 \[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=-\frac {7}{4} \sqrt {2+x+x^2}+\frac {1}{2} x \sqrt {2+x+x^2}-\frac {1}{8} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {7}}\right )+\frac {\arctan \left (\frac {1+2 x}{\sqrt {3} \sqrt {2+x+x^2}}\right )}{\sqrt {3}}-\text {arctanh}\left (\sqrt {2+x+x^2}\right ) \] Output:
-1/8*arcsinh(1/7*(1+2*x)*7^(1/2))-arctanh((x^2+x+2)^(1/2))+1/3*arctan(1/3* (1+2*x)*3^(1/2)/(x^2+x+2)^(1/2))*3^(1/2)-7/4*(x^2+x+2)^(1/2)+1/2*x*(x^2+x+ 2)^(1/2)
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=-\frac {\arctan \left (\frac {2+2 x+2 x^2-(1+2 x) \sqrt {2+x+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\text {arctanh}\left (\sqrt {2+x+x^2}\right )+\frac {1}{8} \left (2 (-7+2 x) \sqrt {2+x+x^2}+\log \left (-1-2 x+2 \sqrt {2+x+x^2}\right )\right ) \] Input:
Integrate[(1 + x^4)/((1 + x + x^2)*Sqrt[2 + x + x^2]),x]
Output:
-(ArcTan[(2 + 2*x + 2*x^2 - (1 + 2*x)*Sqrt[2 + x + x^2])/Sqrt[3]]/Sqrt[3]) - ArcTanh[Sqrt[2 + x + x^2]] + (2*(-7 + 2*x)*Sqrt[2 + x + x^2] + Log[-1 - 2*x + 2*Sqrt[2 + x + x^2]])/8
Time = 0.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \frac {x^4+1}{\left (x^2+x+1\right ) \sqrt {x^2+x+2}} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {x^2}{\sqrt {x^2+x+2}}-\frac {x}{\sqrt {x^2+x+2}}+\frac {x+1}{\left (x^2+x+1\right ) \sqrt {x^2+x+2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {7}}\right )+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3} \sqrt {x^2+x+2}}\right )}{\sqrt {3}}-\text {arctanh}\left (\sqrt {x^2+x+2}\right )+\frac {1}{2} \sqrt {x^2+x+2} x-\frac {7}{4} \sqrt {x^2+x+2}\) |
Input:
Int[(1 + x^4)/((1 + x + x^2)*Sqrt[2 + x + x^2]),x]
Output:
(-7*Sqrt[2 + x + x^2])/4 + (x*Sqrt[2 + x + x^2])/2 - ArcSinh[(1 + 2*x)/Sqr t[7]]/8 + ArcTan[(1 + 2*x)/(Sqrt[3]*Sqrt[2 + x + x^2])]/Sqrt[3] - ArcTanh[ Sqrt[2 + x + x^2]]
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 = 3.48 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {\left (2 x -7\right ) \sqrt {x^{2}+x +2}}{4}-\frac {\operatorname {arcsinh}\left (\frac {2 \left (x +\frac {1}{2}\right ) \sqrt {7}}{7}\right )}{8}-\operatorname {arctanh}\left (\sqrt {x^{2}+x +2}\right )+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3 \sqrt {x^{2}+x +2}}\right ) \sqrt {3}}{3}\) | \(63\) |
default | \(\frac {x \sqrt {x^{2}+x +2}}{2}-\frac {7 \sqrt {x^{2}+x +2}}{4}-\frac {\operatorname {arcsinh}\left (\frac {2 \left (x +\frac {1}{2}\right ) \sqrt {7}}{7}\right )}{8}-\operatorname {arctanh}\left (\sqrt {x^{2}+x +2}\right )+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3 \sqrt {x^{2}+x +2}}\right ) \sqrt {3}}{3}\) | \(69\) |
trager | \(\text {Expression too large to display}\) | \(3736\) |
Input:
int((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4*(2*x-7)*(x^2+x+2)^(1/2)-1/8*arcsinh(2/7*(x+1/2)*7^(1/2))-arctanh((x^2+ x+2)^(1/2))+1/3*arctan(1/3*(1+2*x)*3^(1/2)/(x^2+x+2)^(1/2))*3^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (70) = 140\).
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.69 \[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=\frac {1}{4} \, \sqrt {x^{2} + x + 2} {\left (2 \, x - 7\right )} - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 3\right )} + \frac {2}{3} \, \sqrt {3} \sqrt {x^{2} + x + 2}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )} + \frac {2}{3} \, \sqrt {3} \sqrt {x^{2} + x + 2}\right ) + \frac {1}{2} \, \log \left (2 \, x^{2} - \sqrt {x^{2} + x + 2} {\left (2 \, x + 3\right )} + 4 \, x + 5\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - \sqrt {x^{2} + x + 2} {\left (2 \, x - 1\right )} + 3\right ) + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 2} - 1\right ) \] Input:
integrate((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x, algorithm="fricas")
Output:
1/4*sqrt(x^2 + x + 2)*(2*x - 7) - 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x + 3 ) + 2/3*sqrt(3)*sqrt(x^2 + x + 2)) + 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - 1) + 2/3*sqrt(3)*sqrt(x^2 + x + 2)) + 1/2*log(2*x^2 - sqrt(x^2 + x + 2)* (2*x + 3) + 4*x + 5) - 1/2*log(2*x^2 - sqrt(x^2 + x + 2)*(2*x - 1) + 3) + 1/8*log(-2*x + 2*sqrt(x^2 + x + 2) - 1)
\[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=\int \frac {x^{4} + 1}{\left (x^{2} + x + 1\right ) \sqrt {x^{2} + x + 2}}\, dx \] Input:
integrate((x**4+1)/(x**2+x+1)/(x**2+x+2)**(1/2),x)
Output:
Integral((x**4 + 1)/((x**2 + x + 1)*sqrt(x**2 + x + 2)), x)
\[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{2} + x + 2} {\left (x^{2} + x + 1\right )}} \,d x } \] Input:
integrate((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x, algorithm="maxima")
Output:
integrate((x^4 + 1)/(sqrt(x^2 + x + 2)*(x^2 + x + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.70 \[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=\frac {1}{4} \, \sqrt {x^{2} + x + 2} {\left (2 \, x - 7\right )} - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 2 \, \sqrt {x^{2} + x + 2} + 3\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 2 \, \sqrt {x^{2} + x + 2} - 1\right )}\right ) + \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + x + 2}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} + x + 2} + 3\right ) - \frac {1}{2} \, \log \left ({\left (x - \sqrt {x^{2} + x + 2}\right )}^{2} - x + \sqrt {x^{2} + x + 2} + 1\right ) + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 2} - 1\right ) \] Input:
integrate((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x, algorithm="giac")
Output:
1/4*sqrt(x^2 + x + 2)*(2*x - 7) - 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - 2 *sqrt(x^2 + x + 2) + 3)) + 1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - 2*sqrt(x ^2 + x + 2) - 1)) + 1/2*log((x - sqrt(x^2 + x + 2))^2 + 3*x - 3*sqrt(x^2 + x + 2) + 3) - 1/2*log((x - sqrt(x^2 + x + 2))^2 - x + sqrt(x^2 + x + 2) + 1) + 1/8*log(-2*x + 2*sqrt(x^2 + x + 2) - 1)
Timed out. \[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=\int \frac {x^4+1}{\left (x^2+x+1\right )\,\sqrt {x^2+x+2}} \,d x \] Input:
int((x^4 + 1)/((x + x^2 + 1)*(x + x^2 + 2)^(1/2)),x)
Output:
int((x^4 + 1)/((x + x^2 + 1)*(x + x^2 + 2)^(1/2)), x)
Time = 0.15 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.24 \[ \int \frac {1+x^4}{\left (1+x+x^2\right ) \sqrt {2+x+x^2}} \, dx=\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x^{2}+x +2}+2 x -1}{\sqrt {3}}\right )}{3}-\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x^{2}+x +2}+2 x +3}{\sqrt {3}}\right )}{3}+\frac {\sqrt {x^{2}+x +2}\, x}{2}-\frac {7 \sqrt {x^{2}+x +2}}{4}+\frac {\mathrm {log}\left (\frac {28 \sqrt {x^{2}+x +2}\, x -14 \sqrt {x^{2}+x +2}+28 x^{2}+42}{2 \sqrt {x^{2}+x +2}\, \sqrt {7}+2 \sqrt {7}\, x +\sqrt {7}}\right )}{2}-\frac {\mathrm {log}\left (\frac {28 \sqrt {x^{2}+x +2}\, x +42 \sqrt {x^{2}+x +2}+28 x^{2}+56 x +70}{2 \sqrt {x^{2}+x +2}\, \sqrt {7}+2 \sqrt {7}\, x +\sqrt {7}}\right )}{2}-\frac {\mathrm {log}\left (\frac {2 \sqrt {x^{2}+x +2}+2 x +1}{\sqrt {7}}\right )}{8} \] Input:
int((x^4+1)/(x^2+x+1)/(x^2+x+2)^(1/2),x)
Output:
(8*sqrt(3)*atan((2*sqrt(x**2 + x + 2) + 2*x - 1)/sqrt(3)) - 8*sqrt(3)*atan ((2*sqrt(x**2 + x + 2) + 2*x + 3)/sqrt(3)) + 12*sqrt(x**2 + x + 2)*x - 42* sqrt(x**2 + x + 2) + 12*log((28*sqrt(x**2 + x + 2)*x - 14*sqrt(x**2 + x + 2) + 28*x**2 + 42)/(2*sqrt(x**2 + x + 2)*sqrt(7) + 2*sqrt(7)*x + sqrt(7))) - 12*log((28*sqrt(x**2 + x + 2)*x + 42*sqrt(x**2 + x + 2) + 28*x**2 + 56* x + 70)/(2*sqrt(x**2 + x + 2)*sqrt(7) + 2*sqrt(7)*x + sqrt(7))) - 3*log((2 *sqrt(x**2 + x + 2) + 2*x + 1)/sqrt(7)))/24