Integrand size = 25, antiderivative size = 79 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{3 x^3}+\sqrt {\frac {2}{3}} \text {arctanh}\left (\sqrt {\frac {2}{3}}+\frac {x^6}{\sqrt {6}}+\frac {x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
1/3*(x^6-1)^(1/2)/x^3+1/3*6^(1/2)*arctanh(1/3*6^(1/2)+1/6*x^6*6^(1/2)+1/6* x^3*(x^6-1)^(1/2)*6^(1/2))+1/3*ln(x^3+(x^6-1)^(1/2))
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {1}{3} \left (\frac {\sqrt {-1+x^6}}{x^3}+\sqrt {6} \text {arctanh}\left (\frac {2+x^6+x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )+\log \left (x^3+\sqrt {-1+x^6}\right )\right ) \]
(Sqrt[-1 + x^6]/x^3 + Sqrt[6]*ArcTanh[(2 + x^6 + x^3*Sqrt[-1 + x^6])/Sqrt[ 6]] + Log[x^3 + Sqrt[-1 + x^6]])/3
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.76, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1045, 25, 442, 27, 398, 224, 219, 291, 219}
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.
\(\displaystyle \int \frac {\left (x^6-2\right ) \sqrt {x^6-1}}{x^4 \left (x^6+2\right )} \, dx\) |
\(\Big \downarrow \) 1045 |
\(\displaystyle \frac {1}{3} \int -\frac {\left (2-x^6\right ) \sqrt {x^6-1}}{x^6 \left (x^6+2\right )}dx^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {\left (2-x^6\right ) \sqrt {x^6-1}}{x^6 \left (x^6+2\right )}dx^3\) |
\(\Big \downarrow \) 442 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {x^6-1}}{x^3}-\frac {1}{2} \int \frac {2 \left (4-x^6\right )}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {x^6-1}}{x^3}-\int \frac {4-x^6}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\right )\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{\sqrt {x^6-1}}dx^3-6 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3+\frac {\sqrt {x^6-1}}{x^3}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}-6 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3+\frac {\sqrt {x^6-1}}{x^3}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (-6 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3+\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{3} \left (-6 \int \frac {1}{2-3 x^6}d\frac {x^3}{\sqrt {x^6-1}}+\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {6} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\) |
(Sqrt[-1 + x^6]/x^3 + ArcTanh[x^3/Sqrt[-1 + x^6]] - Sqrt[6]*ArcTanh[(Sqrt[ 3/2]*x^3)/Sqrt[-1 + x^6]])/3
3.11.50.3.1 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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. )*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Si mp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q*(e + f*x^(n/k))^r, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0] && IntegerQ[m]
Time = 2.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {-\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {6}\, x^{3}}{2 \sqrt {x^{6}-1}}\right ) x^{3}+\ln \left (x^{3}+\sqrt {x^{6}-1}\right ) x^{3}+\sqrt {x^{6}-1}}{3 x^{3}}\) | \(54\) |
trager | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\) | \(77\) |
risch | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\) | \(77\) |
1/3*(-6^(1/2)*arctanh(1/2*6^(1/2)*x^3/(x^6-1)^(1/2))*x^3+ln(x^3+(x^6-1)^(1 /2))*x^3+(x^6-1)^(1/2))/x^3
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {\sqrt {3} \sqrt {2} x^{3} \log \left (\frac {25 \, x^{6} - 2 \, \sqrt {3} \sqrt {2} {\left (5 \, x^{6} - 2\right )} - 2 \, \sqrt {x^{6} - 1} {\left (5 \, \sqrt {3} \sqrt {2} x^{3} - 12 \, x^{3}\right )} - 10}{x^{6} + 2}\right ) - 2 \, x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{3} + 2 \, \sqrt {x^{6} - 1}}{6 \, x^{3}} \]
1/6*(sqrt(3)*sqrt(2)*x^3*log((25*x^6 - 2*sqrt(3)*sqrt(2)*(5*x^6 - 2) - 2*s qrt(x^6 - 1)*(5*sqrt(3)*sqrt(2)*x^3 - 12*x^3) - 10)/(x^6 + 2)) - 2*x^3*log (-x^3 + sqrt(x^6 - 1)) + 2*x^3 + 2*sqrt(x^6 - 1))/x^3
\[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{6} - 2\right )}{x^{4} \left (x^{6} + 2\right )}\, dx \]
Integral(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(x**6 - 2)/(x **4*(x**6 + 2)), x)
\[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 2\right )} x^{4}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {\sqrt {6} \log \left (\frac {\sqrt {6} - 2 \, \sqrt {-\frac {1}{x^{6}} + 1}}{\sqrt {6} + 2 \, \sqrt {-\frac {1}{x^{6}} + 1}}\right )}{6 \, \mathrm {sgn}\left (x\right )} + \frac {\log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\left (x\right )} - \frac {\log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\left (x\right )} + \frac {\sqrt {-\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\left (x\right )} \]
1/6*sqrt(6)*log((sqrt(6) - 2*sqrt(-1/x^6 + 1))/(sqrt(6) + 2*sqrt(-1/x^6 + 1)))/sgn(x) + 1/6*log(sqrt(-1/x^6 + 1) + 1)/sgn(x) - 1/6*log(-sqrt(-1/x^6 + 1) + 1)/sgn(x) + 1/3*sqrt(-1/x^6 + 1)/sgn(x)
Timed out. \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\int \frac {\sqrt {x^6-1}\,\left (x^6-2\right )}{x^4\,\left (x^6+2\right )} \,d x \]