\(\int \frac {(-2+x^6) \sqrt {-1+x^6}}{x^4 (2+x^6)} \, dx\) [1050]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{3 x^3}+\sqrt {\frac {2}{3}} \text {arctanh}\left (\sqrt {\frac {2}{3}}+\frac {x^6}{\sqrt {6}}+\frac {x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

[Out]

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))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {589, 594, 537, 223, 212, 385} \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{3 x^3} \]

[In]

Int[((-2 + x^6)*Sqrt[-1 + x^6])/(x^4*(2 + x^6)),x]

[Out]

Sqrt[-1 + x^6]/(3*x^3) + ArcTanh[x^3/Sqrt[-1 + x^6]]/3 - Sqrt[2/3]*ArcTanh[(Sqrt[3/2]*x^3)/Sqrt[-1 + x^6]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 589

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]}, Dist[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] && Inte
gerQ[m]

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\left (-2+x^2\right ) \sqrt {-1+x^2}}{x^2 \left (2+x^2\right )} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} \text {Subst}\left (\int \frac {-8+2 x^2}{\sqrt {-1+x^2} \left (2+x^2\right )} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )-2 \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (2+x^2\right )} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )-2 \text {Subst}\left (\int \frac {1}{2-3 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {1}{3} \left (\frac {\sqrt {-1+x^6}}{x^3}+\sqrt {6} \text {arctanh}\left (\frac {2+x^6+x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )+\log \left (x^3+\sqrt {-1+x^6}\right )\right ) \]

[In]

Integrate[((-2 + x^6)*Sqrt[-1 + x^6])/(x^4*(2 + x^6)),x]

[Out]

(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

Maple [A] (verified)

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	\(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\)	\(77\)

[In]

int((x^6-2)*(x^6-1)^(1/2)/x^4/(x^6+2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {\sqrt {3} \sqrt {2} x^{3} \log \left (\frac {25 \, x^{6} - 2 \, \sqrt {3} \sqrt {2} {\left (5 \, x^{6} - 2\right )} - 2 \, \sqrt {x^{6} - 1} {\left (5 \, \sqrt {3} \sqrt {2} x^{3} - 12 \, x^{3}\right )} - 10}{x^{6} + 2}\right ) - 2 \, x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{3} + 2 \, \sqrt {x^{6} - 1}}{6 \, x^{3}} \]

[In]

integrate((x^6-2)*(x^6-1)^(1/2)/x^4/(x^6+2),x, algorithm="fricas")

[Out]

1/6*(sqrt(3)*sqrt(2)*x^3*log((25*x^6 - 2*sqrt(3)*sqrt(2)*(5*x^6 - 2) - 2*sqrt(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

Sympy [F]

\[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{6} - 2\right )}{x^{4} \left (x^{6} + 2\right )}\, dx \]

[In]

integrate((x**6-2)*(x**6-1)**(1/2)/x**4/(x**6+2),x)

[Out]

Integral(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(x**6 - 2)/(x**4*(x**6 + 2)), x)

Maxima [F]

\[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 2\right )} x^{4}} \,d x } \]

[In]

integrate((x^6-2)*(x^6-1)^(1/2)/x^4/(x^6+2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - 1)*(x^6 - 2)/((x^6 + 2)*x^4), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\frac {\sqrt {6} \log \left (\frac {\sqrt {6} - 2 \, \sqrt {-\frac {1}{x^{6}} + 1}}{\sqrt {6} + 2 \, \sqrt {-\frac {1}{x^{6}} + 1}}\right )}{6 \, \mathrm {sgn}\left (x\right )} + \frac {\log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\left (x\right )} - \frac {\log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\left (x\right )} + \frac {\sqrt {-\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\left (x\right )} \]

[In]

integrate((x^6-2)*(x^6-1)^(1/2)/x^4/(x^6+2),x, algorithm="giac")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx=\int \frac {\sqrt {x^6-1}\,\left (x^6-2\right )}{x^4\,\left (x^6+2\right )} \,d x \]

[In]

int(((x^6 - 1)^(1/2)*(x^6 - 2))/(x^4*(x^6 + 2)),x)

[Out]

int(((x^6 - 1)^(1/2)*(x^6 - 2))/(x^4*(x^6 + 2)), x)