Integrand size = 13, antiderivative size = 56 \[ \int \frac {x^7}{1-x^6} \, dx=-\frac {x^2}{2}+\frac {\arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-x^2\right )+\frac {1}{12} \log \left (1+x^2+x^4\right ) \]
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39 \[ \int \frac {x^7}{1-x^6} \, dx=\frac {1}{12} \left (-6 x^2+2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-2 \log (1-x)-2 \log (1+x)+\log \left (1-x+x^2\right )+\log \left (1+x+x^2\right )\right ) \]
(-6*x^2 + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x )/Sqrt[3]] - 2*Log[1 - x] - 2*Log[1 + x] + Log[1 - x + x^2] + Log[1 + x + x^2])/12
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {807, 843, 750, 16, 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.
\(\displaystyle \int \frac {x^7}{1-x^6} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^6}{1-x^6}dx^2\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{1-x^6}dx^2-x^2\right )\) |
\(\Big \downarrow \) 750 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {1}{1-x^2}dx^2+\frac {1}{3} \int \frac {x^2+2}{x^4+x^2+1}dx^2-x^2\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {x^2+2}{x^4+x^2+1}dx^2-x^2-\frac {1}{3} \log \left (1-x^2\right )\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^4+x^2+1}dx^2+\frac {1}{2} \int \frac {2 x^2+1}{x^4+x^2+1}dx^2\right )-x^2-\frac {1}{3} \log \left (1-x^2\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {2 x^2+1}{x^4+x^2+1}dx^2-3 \int \frac {1}{-x^4-3}d\left (2 x^2+1\right )\right )-x^2-\frac {1}{3} \log \left (1-x^2\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {2 x^2+1}{x^4+x^2+1}dx^2+\sqrt {3} \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )\right )-x^2-\frac {1}{3} \log \left (1-x^2\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (x^4+x^2+1\right )\right )-x^2-\frac {1}{3} \log \left (1-x^2\right )\right )\) |
3.14.45.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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]
Time = 4.43 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-\frac {x^{2}}{2}+\frac {\ln \left (4 x^{4}+4 x^{2}+4\right )}{12}+\frac {\arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (x^{2}-1\right )}{6}\) | \(48\) |
default | \(-\frac {x^{2}}{2}-\frac {\ln \left (-1+x \right )}{6}+\frac {\ln \left (x^{2}+x +1\right )}{12}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{6}\) | \(71\) |
meijerg | \(\frac {\left (-1\right )^{\frac {2}{3}} \left (3 x^{2} \left (-1\right )^{\frac {1}{3}}+\frac {x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{3}}+\left (x^{6}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{2+\left (x^{6}\right )^{\frac {1}{3}}}\right )\right )}{\left (x^{6}\right )^{\frac {1}{3}}}\right )}{6}\) | \(80\) |
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {x^7}{1-x^6} \, dx=-\frac {1}{2} \, x^{2} + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac {1}{6} \, \log \left (x^{2} - 1\right ) \]
-1/2*x^2 + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) + 1/12*log(x^4 + x^ 2 + 1) - 1/6*log(x^2 - 1)
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {x^7}{1-x^6} \, dx=- \frac {x^{2}}{2} - \frac {\log {\left (x^{2} - 1 \right )}}{6} + \frac {\log {\left (x^{4} + x^{2} + 1 \right )}}{12} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \]
-x**2/2 - log(x**2 - 1)/6 + log(x**4 + x**2 + 1)/12 + sqrt(3)*atan(2*sqrt( 3)*x**2/3 + sqrt(3)/3)/6
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {x^7}{1-x^6} \, dx=-\frac {1}{2} \, x^{2} + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac {1}{6} \, \log \left (x^{2} - 1\right ) \]
-1/2*x^2 + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) + 1/12*log(x^4 + x^ 2 + 1) - 1/6*log(x^2 - 1)
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int \frac {x^7}{1-x^6} \, dx=-\frac {1}{2} \, x^{2} + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac {1}{6} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
-1/2*x^2 + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) + 1/12*log(x^4 + x^ 2 + 1) - 1/6*log(abs(x^2 - 1))
Time = 5.46 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {x^7}{1-x^6} \, dx=-\frac {\ln \left (x^2-1\right )}{6}-\frac {x^2}{2}-\ln \left (x^2-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}+\frac {1}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x^2+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}+\frac {1}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]