Integrand size = 16, antiderivative size = 91 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\frac {x^2}{2}-\frac {\arctan \left (x^2\right )}{2}-\frac {\arctan \left (1-\sqrt {2} x\right )}{2 \sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} x\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{1+x^2}\right )}{2 \sqrt {2}}+\log (x)-\frac {1}{4} \log \left (1+x^4\right ) \] Output:
1/2*x^2-1/2*arctan(x^2)+1/4*arctan(-1+x*2^(1/2))*2^(1/2)+1/4*arctan(1+x*2^ (1/2))*2^(1/2)-1/4*arctanh(2^(1/2)*x/(x^2+1))*2^(1/2)+ln(x)-1/4*ln(x^4+1)
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.11 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\frac {1}{8} \left (4 x^2-2 \left (-2+\sqrt {2}\right ) \arctan \left (1-\sqrt {2} x\right )+2 \left (2+\sqrt {2}\right ) \arctan \left (1+\sqrt {2} x\right )+8 \log (x)+\sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )-\sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )-2 \log \left (1+x^4\right )\right ) \] Input:
Integrate[(1 + x^3 + x^6)/(x + x^5),x]
Output:
(4*x^2 - 2*(-2 + Sqrt[2])*ArcTan[1 - Sqrt[2]*x] + 2*(2 + Sqrt[2])*ArcTan[1 + Sqrt[2]*x] + 8*Log[x] + Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - Sqrt[2]*Log[ 1 + Sqrt[2]*x + x^2] - 2*Log[1 + x^4])/8
Time = 0.52 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2026, 2372, 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.
\(\displaystyle \int \frac {x^6+x^3+1}{x^5+x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^6+x^3+1}{x \left (x^4+1\right )}dx\) |
\(\Big \downarrow \) 2372 |
\(\displaystyle \int \left (\frac {x^6+1}{\left (x^4+1\right ) x}+\frac {x^2}{x^4+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (x^2\right )}{2}-\frac {\arctan \left (1-\sqrt {2} x\right )}{2 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {1}{4} \log \left (x^4+1\right )+\frac {x^2}{2}+\frac {\log \left (x^2-\sqrt {2} x+1\right )}{4 \sqrt {2}}-\frac {\log \left (x^2+\sqrt {2} x+1\right )}{4 \sqrt {2}}+\log (x)\) |
Input:
Int[(1 + x^3 + x^6)/(x + x^5),x]
Output:
x^2/2 - ArcTan[x^2]/2 - ArcTan[1 - Sqrt[2]*x]/(2*Sqrt[2]) + ArcTan[1 + Sqr t[2]*x]/(2*Sqrt[2]) + Log[x] + Log[1 - Sqrt[2]*x + x^2]/(4*Sqrt[2]) - Log[ 1 + Sqrt[2]*x + x^2]/(4*Sqrt[2]) - Log[1 + x^4]/4
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {x^{2}}{2}+\ln \left (x \right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}-5 \textit {\_R}^{2}-10 \textit {\_R} +3 x -5\right )\right )}{4}\) | \(54\) |
default | \(\frac {x^{2}}{2}-\frac {\arctan \left (x^{2}\right )}{2}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-x \sqrt {2}+1}{x^{2}+x \sqrt {2}+1}\right )+2 \arctan \left (1+x \sqrt {2}\right )+2 \arctan \left (-1+x \sqrt {2}\right )\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{4}+\ln \left (x \right )\) | \(74\) |
meijerg | \(\frac {x^{2}}{2}-\frac {\arctan \left (x^{2}\right )}{2}+\frac {x^{3} \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}+\sqrt {x^{4}}\right )}{8 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{4 \left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}+\sqrt {x^{4}}\right )}{8 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{4 \left (x^{4}\right )^{\frac {3}{4}}}+\ln \left (x \right )-\frac {\ln \left (x^{4}+1\right )}{4}\) | \(160\) |
Input:
int((x^6+x^3+1)/(x^5+x),x,method=_RETURNVERBOSE)
Output:
1/2*x^2+ln(x)+1/4*sum(_R*ln(-_R^3-5*_R^2-10*_R+3*x-5),_R=RootOf(_Z^4+4*_Z^ 3+8*_Z^2+4*_Z+1))
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{4} \, {\left (\sqrt {2} + 2\right )} \arctan \left (\sqrt {2} x + 1\right ) + \frac {1}{4} \, {\left (\sqrt {2} - 2\right )} \arctan \left (\sqrt {2} x - 1\right ) - \frac {1}{8} \, {\left (\sqrt {2} + 2\right )} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{8} \, {\left (\sqrt {2} - 2\right )} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \log \left (x\right ) \] Input:
integrate((x^6+x^3+1)/(x^5+x),x, algorithm="fricas")
Output:
1/2*x^2 + 1/4*(sqrt(2) + 2)*arctan(sqrt(2)*x + 1) + 1/4*(sqrt(2) - 2)*arct an(sqrt(2)*x - 1) - 1/8*(sqrt(2) + 2)*log(x^2 + sqrt(2)*x + 1) + 1/8*(sqrt (2) - 2)*log(x^2 - sqrt(2)*x + 1) + log(x)
Time = 0.67 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.67 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\frac {x^{2}}{2} + \log {\left (x \right )} + \operatorname {RootSum} {\left (256 t^{4} + 256 t^{3} + 128 t^{2} + 16 t + 1, \left ( t \mapsto t \log {\left (\frac {1792 t^{4}}{73} + \frac {704 t^{3}}{219} - \frac {3152 t^{2}}{219} - \frac {2584 t}{219} + x - \frac {344}{219} \right )} \right )\right )} \] Input:
integrate((x**6+x**3+1)/(x**5+x),x)
Output:
x**2/2 + log(x) + RootSum(256*_t**4 + 256*_t**3 + 128*_t**2 + 16*_t + 1, L ambda(_t, _t*log(1792*_t**4/73 + 704*_t**3/219 - 3152*_t**2/219 - 2584*_t/ 219 + x - 344/219)))
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.09 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} - 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} + 1\right )} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} - 1\right )} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {1}{2} \, x^{2} + \log \left (x\right ) \] Input:
integrate((x^6+x^3+1)/(x^5+x),x, algorithm="maxima")
Output:
1/4*sqrt(2)*(sqrt(2) + 1)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) - 1/4*sqrt(2 )*(sqrt(2) - 1)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) - 1/8*sqrt(2)*(sqrt(2) + 1)*log(x^2 + sqrt(2)*x + 1) - 1/8*sqrt(2)*(sqrt(2) - 1)*log(x^2 - sqrt( 2)*x + 1) + 1/2*x^2 + log(x)
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{4} \, {\left (\sqrt {2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{4} \, {\left (\sqrt {2} - 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) - \frac {1}{4} \, \log \left (x^{4} + 1\right ) + \log \left ({\left | x \right |}\right ) \] Input:
integrate((x^6+x^3+1)/(x^5+x),x, algorithm="giac")
Output:
1/2*x^2 + 1/4*(sqrt(2) + 2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/4*(sqr t(2) - 2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) - 1/8*sqrt(2)*log(x^2 + sqrt (2)*x + 1) + 1/8*sqrt(2)*log(x^2 - sqrt(2)*x + 1) - 1/4*log(x^4 + 1) + log (abs(x))
Time = 22.49 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.87 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=\ln \left (x\right )+\left (\sum _{k=1}^4\ln \left (\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )\,\left (8\,\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )+x+\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )\,x\,96+{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )}^2\,x\,240+{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )}^3\,x\,320-16\,{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )}^2+8\right )\right )\,\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}+\frac {z}{16}+\frac {1}{256},z,k\right )\right )+\frac {x^2}{2} \] Input:
int((x^3 + x^6 + 1)/(x + x^5),x)
Output:
log(x) + symsum(log(root(z^4 + z^3 + z^2/2 + z/16 + 1/256, z, k)*(8*root(z ^4 + z^3 + z^2/2 + z/16 + 1/256, z, k) + x + 96*root(z^4 + z^3 + z^2/2 + z /16 + 1/256, z, k)*x + 240*root(z^4 + z^3 + z^2/2 + z/16 + 1/256, z, k)^2* x + 320*root(z^4 + z^3 + z^2/2 + z/16 + 1/256, z, k)^3*x - 16*root(z^4 + z ^3 + z^2/2 + z/16 + 1/256, z, k)^2 + 8))*root(z^4 + z^3 + z^2/2 + z/16 + 1 /256, z, k), k, 1, 4) + x^2/2
Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.34 \[ \int \frac {1+x^3+x^6}{x+x^5} \, dx=-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right )}{4}+\frac {\mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right )}{2}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right )}{4}+\frac {\mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right )}{2}+\frac {\sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, x +x^{2}+1\right )}{8}-\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, x +x^{2}+1\right )}{8}-\frac {\mathrm {log}\left (-\sqrt {2}\, x +x^{2}+1\right )}{4}-\frac {\mathrm {log}\left (\sqrt {2}\, x +x^{2}+1\right )}{4}+\mathrm {log}\left (x \right )+\frac {x^{2}}{2} \] Input:
int((x^6+x^3+1)/(x^5+x),x)
Output:
( - 2*sqrt(2)*atan((sqrt(2) - 2*x)/sqrt(2)) + 4*atan((sqrt(2) - 2*x)/sqrt( 2)) + 2*sqrt(2)*atan((sqrt(2) + 2*x)/sqrt(2)) + 4*atan((sqrt(2) + 2*x)/sqr t(2)) + sqrt(2)*log( - sqrt(2)*x + x**2 + 1) - sqrt(2)*log(sqrt(2)*x + x** 2 + 1) - 2*log( - sqrt(2)*x + x**2 + 1) - 2*log(sqrt(2)*x + x**2 + 1) + 8* log(x) + 4*x**2)/8