Integrand size = 25, antiderivative size = 63 \[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\sqrt {\frac {2}{3}} \text {arctanh}\left (\sqrt {\frac {2}{3}}+\frac {x^6}{\sqrt {6}}+\frac {x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \] Output:
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 = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {1}{3} \left (-\sqrt {6} \text {arctanh}\left (\frac {2+x^6-x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )-\log \left (-x^3+\sqrt {-1+x^6}\right )\right ) \] Input:
Integrate[(x^2*(-4 + x^6))/(Sqrt[-1 + x^6]*(2 + x^6)),x]
Output:
(-(Sqrt[6]*ArcTanh[(2 + x^6 - x^3*Sqrt[-1 + x^6])/Sqrt[6]]) - Log[-x^3 + S qrt[-1 + x^6]])/3
Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1045, 25, 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.
\(\displaystyle \int \frac {x^2 \left (x^6-4\right )}{\sqrt {x^6-1} \left (x^6+2\right )} \, dx\) |
\(\Big \downarrow \) 1045 |
\(\displaystyle \frac {1}{3} \int -\frac {4-x^6}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {4-x^6}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{\sqrt {x^6-1}}dx^3-6 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}-6 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-6 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^3\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-6 \int \frac {1}{2-3 x^6}d\frac {x^3}{\sqrt {x^6-1}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {6} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right )\right )\) |
Input:
Int[(x^2*(-4 + x^6))/(Sqrt[-1 + x^6]*(2 + x^6)),x]
Output:
(ArcTanh[x^3/Sqrt[-1 + x^6]] - Sqrt[6]*ArcTanh[(Sqrt[3/2]*x^3)/Sqrt[-1 + x ^6]])/3
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[(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 = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59
method | result | size |
pseudoelliptic | \(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {6}\, x^{3}}{2 \sqrt {x^{6}-1}}\right )}{3}\) | \(37\) |
trager | \(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{6}-12 x^{3} \sqrt {x^{6}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\) | \(66\) |
Input:
int(x^2*(x^6-4)/(x^6-1)^(1/2)/(x^6+2),x,method=_RETURNVERBOSE)
Output:
1/3*ln(x^3+(x^6-1)^(1/2))-1/3*6^(1/2)*arctanh(1/2*6^(1/2)*x^3/(x^6-1)^(1/2 ))
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.16 \[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {1}{2} \, \sqrt {\frac {2}{3}} \log \left (\frac {25 \, x^{6} - 6 \, \sqrt {\frac {2}{3}} {\left (5 \, x^{6} - 2\right )} - 6 \, \sqrt {x^{6} - 1} {\left (5 \, \sqrt {\frac {2}{3}} x^{3} - 4 \, x^{3}\right )} - 10}{x^{6} + 2}\right ) - \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \] Input:
integrate(x^2*(x^6-4)/(x^6-1)^(1/2)/(x^6+2),x, algorithm="fricas")
Output:
1/2*sqrt(2/3)*log((25*x^6 - 6*sqrt(2/3)*(5*x^6 - 2) - 6*sqrt(x^6 - 1)*(5*s qrt(2/3)*x^3 - 4*x^3) - 10)/(x^6 + 2)) - 1/3*log(-x^3 + sqrt(x^6 - 1))
\[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\int \frac {x^{2} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{6} + 2\right )}\, dx \] Input:
integrate(x**2*(x**6-4)/(x**6-1)**(1/2)/(x**6+2),x)
Output:
Integral(x**2*(x**3 - 2)*(x**3 + 2)/(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*( x**2 + x + 1))*(x**6 + 2)), x)
\[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 4\right )} x^{2}}{{\left (x^{6} + 2\right )} \sqrt {x^{6} - 1}} \,d x } \] Input:
integrate(x^2*(x^6-4)/(x^6-1)^(1/2)/(x^6+2),x, algorithm="maxima")
Output:
integrate((x^6 - 4)*x^2/((x^6 + 2)*sqrt(x^6 - 1)), x)
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {1}{6} \, \sqrt {6} \log \left (\frac {{\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2} - 2 \, \sqrt {6} + 5}{{\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2} + 2 \, \sqrt {6} + 5}\right ) - \frac {1}{6} \, \log \left ({\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2}\right ) \] Input:
integrate(x^2*(x^6-4)/(x^6-1)^(1/2)/(x^6+2),x, algorithm="giac")
Output:
1/6*sqrt(6)*log(((x^3 - sqrt(x^6 - 1))^2 - 2*sqrt(6) + 5)/((x^3 - sqrt(x^6 - 1))^2 + 2*sqrt(6) + 5)) - 1/6*log((x^3 - sqrt(x^6 - 1))^2)
Timed out. \[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\int \frac {x^2\,\left (x^6-4\right )}{\sqrt {x^6-1}\,\left (x^6+2\right )} \,d x \] Input:
int((x^2*(x^6 - 4))/((x^6 - 1)^(1/2)*(x^6 + 2)),x)
Output:
int((x^2*(x^6 - 4))/((x^6 - 1)^(1/2)*(x^6 + 2)), x)
\[ \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx=-6 \left (\int \frac {\sqrt {x^{6}-1}\, x^{2}}{x^{12}+x^{6}-2}d x \right )+\frac {\mathrm {log}\left (\sqrt {x^{6}-1}+x^{3}\right )}{6}-\frac {\mathrm {log}\left (\sqrt {x^{6}-1}-x^{3}\right )}{6} \] Input:
int(x^2*(x^6-4)/(x^6-1)^(1/2)/(x^6+2),x)
Output:
( - 36*int((sqrt(x**6 - 1)*x**2)/(x**12 + x**6 - 2),x) + log(sqrt(x**6 - 1 ) + x**3) - log(sqrt(x**6 - 1) - x**3))/6