Integrand size = 16, antiderivative size = 73 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{6 x^6}-\frac {1}{2 x^2}+\frac {1}{4} \arctan \left (\sqrt {3}-2 x^2\right )-\frac {1}{4} \arctan \left (\sqrt {3}+2 x^2\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} x^2}{1+x^4}\right )}{4 \sqrt {3}} \] Output:
-1/6/x^6-1/2/x^2-1/4*arctan(2*x^2-3^(1/2))-1/4*arctan(3^(1/2)+2*x^2)+1/12* arctanh(3^(1/2)*x^2/(x^4+1))*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{6 x^6}-\frac {1}{2 x^2}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{-1+2 \text {$\#$1}^4}\&\right ] \] Input:
Integrate[1/(x^7*(1 - x^4 + x^8)),x]
Output:
-1/6*1/x^6 - 1/(2*x^2) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^2)/(- 1 + 2*#1^4) & ]/4
Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1695, 1443, 27, 1604, 1447, 1475, 1083, 217, 1478, 25, 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.
\(\displaystyle \int \frac {1}{x^7 \left (x^8-x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 1695 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^8 \left (x^8-x^4+1\right )}dx^2\) |
\(\Big \downarrow \) 1443 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {3 \left (1-x^4\right )}{x^4 \left (x^8-x^4+1\right )}dx^2-\frac {1}{3 x^6}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1-x^4}{x^4 \left (x^8-x^4+1\right )}dx^2-\frac {1}{3 x^6}\right )\) |
\(\Big \downarrow \) 1604 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {x^4}{x^8-x^4+1}dx^2-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 1447 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1-x^4}{x^8-x^4+1}dx^2-\frac {1}{2} \int \frac {x^4+1}{x^8-x^4+1}dx^2-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 1475 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{x^4-\sqrt {3} x^2+1}dx^2-\frac {1}{2} \int \frac {1}{x^4+\sqrt {3} x^2+1}dx^2\right )+\frac {1}{2} \int \frac {1-x^4}{x^8-x^4+1}dx^2-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\int \frac {1}{-x^4-1}d\left (2 x^2-\sqrt {3}\right )+\int \frac {1}{-x^4-1}d\left (2 x^2+\sqrt {3}\right )\right )+\frac {1}{2} \int \frac {1-x^4}{x^8-x^4+1}dx^2-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1-x^4}{x^8-x^4+1}dx^2+\frac {1}{2} \left (\arctan \left (\sqrt {3}-2 x^2\right )-\arctan \left (2 x^2+\sqrt {3}\right )\right )-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 1478 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx^2}{2 \sqrt {3}}-\frac {\int -\frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx^2}{2 \sqrt {3}}\right )+\frac {1}{2} \left (\arctan \left (\sqrt {3}-2 x^2\right )-\arctan \left (2 x^2+\sqrt {3}\right )\right )-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx^2}{2 \sqrt {3}}+\frac {\int \frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx^2}{2 \sqrt {3}}\right )+\frac {1}{2} \left (\arctan \left (\sqrt {3}-2 x^2\right )-\arctan \left (2 x^2+\sqrt {3}\right )\right )-\frac {1}{3 x^6}-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\arctan \left (\sqrt {3}-2 x^2\right )-\arctan \left (2 x^2+\sqrt {3}\right )\right )-\frac {1}{3 x^6}-\frac {1}{x^2}+\frac {1}{2} \left (\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{2 \sqrt {3}}-\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{2 \sqrt {3}}\right )\right )\) |
Input:
Int[1/(x^7*(1 - x^4 + x^8)),x]
Output:
(-1/3*1/x^6 - x^(-2) + (ArcTan[Sqrt[3] - 2*x^2] - ArcTan[Sqrt[3] + 2*x^2]) /2 + (-1/2*Log[1 - Sqrt[3]*x^2 + x^4]/Sqrt[3] + Log[1 + Sqrt[3]*x^2 + x^4] /(2*Sqrt[3]))/2)/2
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[(-(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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim p[1/(a*d^2*(m + 1)) Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 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]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 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]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) /(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (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) + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {-\frac {x^{4}}{2}-\frac {1}{6}}{x^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-6 \textit {\_R}^{3}+x^{2}-\textit {\_R} \right )\right )}{4}\) | \(46\) |
default | \(\frac {\sqrt {3}\, \left (\frac {\ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{2}-\sqrt {3}\, \arctan \left (2 x^{2}+\sqrt {3}\right )\right )}{12}+\frac {\sqrt {3}\, \left (-\frac {\ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{2}-\sqrt {3}\, \arctan \left (2 x^{2}-\sqrt {3}\right )\right )}{12}-\frac {1}{6 x^{6}}-\frac {1}{2 x^{2}}\) | \(87\) |
Input:
int(1/x^7/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
(-1/2*x^4-1/6)/x^6+1/4*sum(_R*ln(-6*_R^3+x^2-_R),_R=RootOf(9*_Z^4+3*_Z^2+1 ))
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=\frac {\sqrt {3} x^{6} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \sqrt {3} x^{6} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) - 6 \, x^{6} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + 6 \, x^{6} \arctan \left (-2 \, x^{2} + \sqrt {3}\right ) - 12 \, x^{4} - 4}{24 \, x^{6}} \] Input:
integrate(1/x^7/(x^8-x^4+1),x, algorithm="fricas")
Output:
1/24*(sqrt(3)*x^6*log(x^4 + sqrt(3)*x^2 + 1) - sqrt(3)*x^6*log(x^4 - sqrt( 3)*x^2 + 1) - 6*x^6*arctan(2*x^2 + sqrt(3)) + 6*x^6*arctan(-2*x^2 + sqrt(3 )) - 12*x^4 - 4)/x^6
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=- \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{24} - \frac {\operatorname {atan}{\left (2 x^{2} - \sqrt {3} \right )}}{4} - \frac {\operatorname {atan}{\left (2 x^{2} + \sqrt {3} \right )}}{4} + \frac {- 3 x^{4} - 1}{6 x^{6}} \] Input:
integrate(1/x**7/(x**8-x**4+1),x)
Output:
-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/24 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/24 - atan(2*x**2 - sqrt(3))/4 - atan(2*x**2 + sqrt(3))/4 + (-3*x**4 - 1)/(6*x**6)
\[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - x^{4} + 1\right )} x^{7}} \,d x } \] Input:
integrate(1/x^7/(x^8-x^4+1),x, algorithm="maxima")
Output:
-1/6*(3*x^4 + 1)/x^6 - integrate(x^5/(x^8 - x^4 + 1), x)
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {3} x^{4} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) + \frac {1}{12} \, \sqrt {3} x^{4} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) - \frac {3 \, x^{4} + 1}{6 \, x^{6}} \] Input:
integrate(1/x^7/(x^8-x^4+1),x, algorithm="giac")
Output:
-1/12*sqrt(3)*x^4*log(x^4 + sqrt(3)*x^2 + 1) + 1/12*sqrt(3)*x^4*log(x^4 - sqrt(3)*x^2 + 1) - 1/6*(3*x^4 + 1)/x^6
Time = 19.50 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=\mathrm {atan}\left (\frac {2\,x^2}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\mathrm {atan}\left (\frac {2\,x^2}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {\frac {x^4}{2}+\frac {1}{6}}{x^6} \] Input:
int(1/(x^7*(x^8 - x^4 + 1)),x)
Output:
atan((2*x^2)/(3^(1/2)*1i - 1))*((3^(1/2)*1i)/12 + 1/4) + atan((2*x^2)/(3^( 1/2)*1i + 1))*((3^(1/2)*1i)/12 - 1/4) - (x^4/2 + 1/6)/x^6
Time = 0.18 (sec) , antiderivative size = 381, normalized size of antiderivative = 5.22 \[ \int \frac {1}{x^7 \left (1-x^4+x^8\right )} \, dx=\frac {3 \sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{6}+3 \sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{6}+\sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{6}+\sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{6}-\sqrt {3}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{6}-\sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{6}-12 x^{4}-4}{24 x^{6}} \] Input:
int(1/x^7/(x^8-x^4+1),x)
Output:
(3*sqrt( - sqrt(3) + 2)*sqrt(6)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**6 + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((sqrt(6) + sqrt( 2) - 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**6 + 3*sqrt( - sqrt(3) + 2)*sqrt(6)* atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**6 + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2 )))*x**6 + 3*sqrt( - sqrt(3) + 2)*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4 *x)/(sqrt(6) + sqrt(2)))*x**6 + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*sqr t( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2)))*x**6 + 3*sqrt( - sqrt(3) + 2 )*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x**6 + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6 ) + sqrt(2)))*x**6 + sqrt(3)*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x** 6 + sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x**6 - sqrt(3)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2)*x**6 - sqrt(3)*log((sqrt(6)*x + sqr t(2)*x + 2*x**2 + 2)/2)*x**6 - 12*x**4 - 4)/(24*x**6)