Integrand size = 13, antiderivative size = 97 \[ \int \frac {x^2}{1-x^8} \, dx=-\frac {\arctan (x)}{4}-\frac {\arctan \left (1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\text {arctanh}(x)}{4}+\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}} \]
-1/4*arctan(x)+1/4*arctanh(x)+1/8*arctan(-1+x*2^(1/2))*2^(1/2)+1/8*arctan( 1+x*2^(1/2))*2^(1/2)+1/16*ln(1+x^2-x*2^(1/2))*2^(1/2)-1/16*ln(1+x^2+x*2^(1 /2))*2^(1/2)
Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int \frac {x^2}{1-x^8} \, dx=\frac {1}{16} \left (-4 \arctan (x)-2 \sqrt {2} \arctan \left (1-\sqrt {2} x\right )+2 \sqrt {2} \arctan \left (1+\sqrt {2} x\right )-2 \log (1-x)+2 \log (1+x)+\sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )-\sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )\right ) \]
(-4*ArcTan[x] - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 2*Sqrt[2]*ArcTan[1 + Sqr t[2]*x] - 2*Log[1 - x] + 2*Log[1 + x] + Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/16
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {829, 826, 827, 216, 219, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {x^2}{1-x^8} \, dx\) |
| \(\Big \downarrow \) 829 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{1-x^4}dx+\frac {1}{2} \int \frac {x^2}{x^4+1}dx\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{1-x^4}dx+\frac {1}{2} \left (\frac {1}{2} \int \frac {x^2+1}{x^4+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-x^2}dx-\frac {1}{2} \int \frac {1}{x^2+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} \int \frac {x^2+1}{x^4+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-x^2}dx-\frac {\arctan (x)}{2}\right )+\frac {1}{2} \left (\frac {1}{2} \int \frac {x^2+1}{x^4+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {x^2+1}{x^4+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {2} x+1}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {2} x+1}dx\right )-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\left (1-\sqrt {2} x\right )^2-1}d\left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\left (\sqrt {2} x+1\right )^2-1}d\left (\sqrt {2} x+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x^2}{x^4+1}dx\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 x}{x^2-\sqrt {2} x+1}dx}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} x+1\right )}{x^2+\sqrt {2} x+1}dx}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 x}{x^2-\sqrt {2} x+1}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} x+1\right )}{x^2+\sqrt {2} x+1}dx}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 x}{x^2-\sqrt {2} x+1}dx}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} x+1}{x^2+\sqrt {2} x+1}dx\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\text {arctanh}(x)}{2}-\frac {\arctan (x)}{2}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} x\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^2+\sqrt {2} x+1\right )}{2 \sqrt {2}}\right )\right )\) |
(-1/2*ArcTan[x] + ArcTanh[x]/2)/2 + ((-(ArcTan[1 - Sqrt[2]*x]/Sqrt[2]) + A rcTan[1 + Sqrt[2]*x]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*x + x^2]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*x + x^2]/(2*Sqrt[2]))/2)/2
3.15.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[x^m/(r + s*x^ (n/2)), x], x] + Simp[r/(2*a) Int[x^m/(r - s*x^(n/2)), x], x]] /; FreeQ[{ a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && !GtQ[a/b, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3}+x \right )\right )}{8}-\frac {\ln \left (-1+x \right )}{8}-\frac {\arctan \left (x \right )}{4}+\frac {\ln \left (1+x \right )}{8}\) | \(37\) |
| default | \(-\frac {\ln \left (-1+x \right )}{8}+\frac {\ln \left (1+x \right )}{8}-\frac {\arctan \left (x \right )}{4}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x^{2}-\sqrt {2}\, x}{1+x^{2}+\sqrt {2}\, x}\right )+2 \arctan \left (\sqrt {2}\, x +1\right )+2 \arctan \left (\sqrt {2}\, x -1\right )\right )}{16}\) | \(69\) |
| meijerg | \(-\frac {x^{3} \left (\ln \left (1-\left (x^{8}\right )^{\frac {1}{8}}\right )-\ln \left (1+\left (x^{8}\right )^{\frac {1}{8}}\right )-\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{2}-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}{2-\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}\right )+2 \arctan \left (\left (x^{8}\right )^{\frac {1}{8}}\right )+\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{2}-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}{2+\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}\right )\right )}{8 \left (x^{8}\right )^{\frac {3}{8}}}\) | \(145\) |
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{1-x^8} \, dx=\left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (2 \, x + \left (i + 1\right ) \, \sqrt {2}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (2 \, x - \left (i - 1\right ) \, \sqrt {2}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (2 \, x + \left (i - 1\right ) \, \sqrt {2}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (2 \, x - \left (i + 1\right ) \, \sqrt {2}\right ) - \frac {1}{4} \, \arctan \left (x\right ) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \]
(1/16*I - 1/16)*sqrt(2)*log(2*x + (I + 1)*sqrt(2)) - (1/16*I + 1/16)*sqrt( 2)*log(2*x - (I - 1)*sqrt(2)) + (1/16*I + 1/16)*sqrt(2)*log(2*x + (I - 1)* sqrt(2)) - (1/16*I - 1/16)*sqrt(2)*log(2*x - (I + 1)*sqrt(2)) - 1/4*arctan (x) + 1/8*log(x + 1) - 1/8*log(x - 1)
Result contains complex when optimal does not.
Time = 127.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.47 \[ \int \frac {x^2}{1-x^8} \, dx=- \frac {\log {\left (x - 1 \right )}}{8} + \frac {\log {\left (x + 1 \right )}}{8} + \frac {i \log {\left (x - i \right )}}{8} - \frac {i \log {\left (x + i \right )}}{8} - \operatorname {RootSum} {\left (4096 t^{4} + 1, \left ( t \mapsto t \log {\left (- 512 t^{3} + x \right )} \right )\right )} \]
-log(x - 1)/8 + log(x + 1)/8 + I*log(x - I)/8 - I*log(x + I)/8 - RootSum(4 096*_t**4 + 1, Lambda(_t, _t*log(-512*_t**3 + x)))
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{1-x^8} \, dx=\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) - \frac {1}{4} \, \arctan \left (x\right ) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \]
1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/8*sqrt(2)*arctan(1/2*s qrt(2)*(2*x - sqrt(2))) - 1/16*sqrt(2)*log(x^2 + sqrt(2)*x + 1) + 1/16*sqr t(2)*log(x^2 - sqrt(2)*x + 1) - 1/4*arctan(x) + 1/8*log(x + 1) - 1/8*log(x - 1)
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{1-x^8} \, dx=\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) - \frac {1}{4} \, \arctan \left (x\right ) + \frac {1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \]
1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/8*sqrt(2)*arctan(1/2*s qrt(2)*(2*x - sqrt(2))) - 1/16*sqrt(2)*log(x^2 + sqrt(2)*x + 1) + 1/16*sqr t(2)*log(x^2 - sqrt(2)*x + 1) - 1/4*arctan(x) + 1/8*log(abs(x + 1)) - 1/8* log(abs(x - 1))
Time = 6.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.46 \[ \int \frac {x^2}{1-x^8} \, dx=-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\mathrm {atan}\left (x\right )}{4}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right ) \]