Integrand size = 10, antiderivative size = 39 \[ \int \frac {1}{x+x^6+x^{11}} \, dx=-\frac {\arctan \left (\frac {1+2 x^5}{\sqrt {3}}\right )}{5 \sqrt {3}}+\log (x)-\frac {1}{10} \log \left (1+x^5+x^{10}\right ) \] Output:
-1/15*arctan(1/3*(2*x^5+1)*3^(1/2))*3^(1/2)+ln(x)-1/10*ln(x^10+x^5+1)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 197, normalized size of antiderivative = 5.05 \[ \int \frac {1}{x+x^6+x^{11}} \, dx=\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{5 \sqrt {3}}+\log (x)-\frac {1}{10} \log \left (1+x+x^2\right )-\frac {1}{5} \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^3-\text {$\#$1}^4+\text {$\#$1}^5-\text {$\#$1}^7+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1}) \text {$\#$1}+2 \log (x-\text {$\#$1}) \text {$\#$1}^2-\log (x-\text {$\#$1}) \text {$\#$1}^3+3 \log (x-\text {$\#$1}) \text {$\#$1}^4-\log (x-\text {$\#$1}) \text {$\#$1}^5-3 \log (x-\text {$\#$1}) \text {$\#$1}^6+4 \log (x-\text {$\#$1}) \text {$\#$1}^7}{-1+3 \text {$\#$1}^2-4 \text {$\#$1}^3+5 \text {$\#$1}^4-7 \text {$\#$1}^6+8 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(x + x^6 + x^11)^(-1),x]
Output:
ArcTan[(1 + 2*x)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x + x^2]/10 - Roo tSum[1 - #1 + #1^3 - #1^4 + #1^5 - #1^7 + #1^8 & , (-(Log[x - #1]*#1) + 2* Log[x - #1]*#1^2 - Log[x - #1]*#1^3 + 3*Log[x - #1]*#1^4 - Log[x - #1]*#1^ 5 - 3*Log[x - #1]*#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^ 4 - 7*#1^6 + 8*#1^7) & ]/5
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {1949, 1693, 1144, 25, 1142, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {1}{x^{11}+x^6+x} \, dx\) |
\(\Big \downarrow \) 1949 |
\(\displaystyle \int \frac {1}{x \left (x^{10}+x^5+1\right )}dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle \frac {1}{5} \int \frac {1}{x^5 \left (x^{10}+x^5+1\right )}dx^5\) |
\(\Big \downarrow \) 1144 |
\(\displaystyle \frac {1}{5} \left (\int -\frac {x^5+1}{x^{10}+x^5+1}dx^5+\log \left (x^5\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{5} \left (\log \left (x^5\right )-\int \frac {x^5+1}{x^{10}+x^5+1}dx^5\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{5} \left (-\frac {1}{2} \int \frac {1}{x^{10}+x^5+1}dx^5-\frac {1}{2} \int \frac {2 x^5+1}{x^{10}+x^5+1}dx^5+\log \left (x^5\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{5} \left (\int \frac {1}{-x^{10}-3}d\left (2 x^5+1\right )-\frac {1}{2} \int \frac {2 x^5+1}{x^{10}+x^5+1}dx^5+\log \left (x^5\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{5} \left (-\frac {1}{2} \int \frac {2 x^5+1}{x^{10}+x^5+1}dx^5-\frac {\arctan \left (\frac {2 x^5+1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (x^5\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{5} \left (-\frac {\arctan \left (\frac {2 x^5+1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (x^5\right )-\frac {1}{2} \log \left (x^{10}+x^5+1\right )\right )\) |
Input:
Int[(x + x^6 + x^11)^(-1),x]
Output:
(-(ArcTan[(1 + 2*x^5)/Sqrt[3]]/Sqrt[3]) + Log[x^5] - Log[1 + x^5 + x^10]/2 )/5
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[e*(Log[RemoveContent[d + e*x, x]]/(c*d^2 - b*d*e + a*e^2)), x] + S imp[1/(c*d^2 - b*d*e + a*e^2) Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Int[((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol ] :> Int[x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && IntegerQ[p]
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\ln \left (x \right )-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{5}+\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{15}-\frac {\ln \left (x^{10}+x^{5}+1\right )}{10}\) | \(31\) |
default | \(-\frac {\ln \left (x^{2}+x +1\right )}{10}+\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{15}-\frac {\left (\frac {i \sqrt {3}}{6}+\frac {1}{2}\right ) \ln \left (2 x^{4}+\left (-1+i \sqrt {3}\right ) x^{3}+\left (-1-i \sqrt {3}\right ) x^{2}+2 x -1+i \sqrt {3}\right )}{5}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{6}\right ) \ln \left (2 x^{4}+\left (-1-i \sqrt {3}\right ) x^{3}+\left (-1+i \sqrt {3}\right ) x^{2}+2 x -1-i \sqrt {3}\right )}{5}+\ln \left (x \right )\) | \(131\) |
Input:
int(1/(x^11+x^6+x),x,method=_RETURNVERBOSE)
Output:
ln(x)-1/15*3^(1/2)*arctan(2/3*(x^5+1/2)*3^(1/2))-1/10*ln(x^10+x^5+1)
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x+x^6+x^{11}} \, dx=-\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) + \log \left (x\right ) \] Input:
integrate(1/(x^11+x^6+x),x, algorithm="fricas")
Output:
-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/10*log(x^10 + x^5 + 1) + log(x)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x+x^6+x^{11}} \, dx=\log {\left (x \right )} - \frac {\log {\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{5}}{3} + \frac {\sqrt {3}}{3} \right )}}{15} \] Input:
integrate(1/(x**11+x**6+x),x)
Output:
log(x) - log(x**10 + x**5 + 1)/10 - sqrt(3)*atan(2*sqrt(3)*x**5/3 + sqrt(3 )/3)/15
\[ \int \frac {1}{x+x^6+x^{11}} \, dx=\int { \frac {1}{x^{11} + x^{6} + x} \,d x } \] Input:
integrate(1/(x^11+x^6+x),x, algorithm="maxima")
Output:
1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/5*integrate((4*x^7 - 3*x^6 - x^5 + 3*x^4 - x^3 + 2*x^2 - x)/(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1), x) - 1/10*log(x^2 + x + 1) + log(x)
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x+x^6+x^{11}} \, dx=-\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) + \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/(x^11+x^6+x),x, algorithm="giac")
Output:
-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/10*log(x^10 + x^5 + 1) + log(abs(x))
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x+x^6+x^{11}} \, dx=\ln \left (x\right )-\frac {\ln \left (x^{10}+x^5+1\right )}{10}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^5}{3}+\frac {\sqrt {3}}{3}\right )}{15} \] Input:
int(1/(x + x^6 + x^11),x)
Output:
log(x) - log(x^5 + x^10 + 1)/10 - (3^(1/2)*atan(3^(1/2)/3 + (2*3^(1/2)*x^5 )/3))/15
\[ \int \frac {1}{x+x^6+x^{11}} \, dx=\int \frac {1}{x^{11}+x^{6}+x}d x \] Input:
int(1/(x^11+x^6+x),x)
Output:
int(1/(x**11 + x**6 + x),x)