Integrand size = 25, antiderivative size = 53 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{2} \arctan \left (\sqrt {-1+x^6}\right )+\frac {\arctan \left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \]
1/6*(x^6-1)^(1/2)/x^6-1/2*arctan((x^6-1)^(1/2))+1/3*arctan(1/3*(x^6-1)^(1/ 2)*3^(1/2))*3^(1/2)
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{2} \arctan \left (\sqrt {-1+x^6}\right )+\frac {\arctan \left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \]
Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1043, 25, 166, 27, 174, 73, 216}
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.
\(\displaystyle \int \frac {\left (x^6-2\right ) \sqrt {x^6-1}}{x^7 \left (x^6+2\right )} \, dx\) |
\(\Big \downarrow \) 1043 |
\(\displaystyle \frac {1}{6} \int -\frac {\left (2-x^6\right ) \sqrt {x^6-1}}{x^{12} \left (x^6+2\right )}dx^6\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{6} \int \frac {\left (2-x^6\right ) \sqrt {x^6-1}}{x^{12} \left (x^6+2\right )}dx^6\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {1}{6} \left (\frac {\sqrt {x^6-1}}{x^6}-\frac {1}{2} \int \frac {3 \left (2-x^6\right )}{x^6 \sqrt {x^6-1} \left (x^6+2\right )}dx^6\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {\sqrt {x^6-1}}{x^6}-\frac {3}{2} \int \frac {2-x^6}{x^6 \sqrt {x^6-1} \left (x^6+2\right )}dx^6\right )\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{6} \left (\frac {\sqrt {x^6-1}}{x^6}-\frac {3}{2} \left (\int \frac {1}{x^6 \sqrt {x^6-1}}dx^6-2 \int \frac {1}{\sqrt {x^6-1} \left (x^6+2\right )}dx^6\right )\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} \left (\frac {\sqrt {x^6-1}}{x^6}-\frac {3}{2} \left (2 \int \frac {1}{x^{12}+1}d\sqrt {x^6-1}-4 \int \frac {1}{x^{12}+3}d\sqrt {x^6-1}\right )\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{6} \left (\frac {\sqrt {x^6-1}}{x^6}-\frac {3}{2} \left (2 \arctan \left (\sqrt {x^6-1}\right )-\frac {4 \arctan \left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )}{\sqrt {3}}\right )\right )\) |
(Sqrt[-1 + x^6]/x^6 - (3*(2*ArcTan[Sqrt[-1 + x^6]] - (4*ArcTan[Sqrt[-1 + x ^6]/Sqrt[3]])/Sqrt[3]))/2)/6
3.7.79.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. )*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simp lify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] / ; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/ n]]
Time = 2.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {x^{6}-1}\, \sqrt {3}}{3}\right ) \sqrt {3}\, x^{6}+3 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{6}+\sqrt {x^{6}-1}}{6 x^{6}}\) | \(48\) |
trager | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+6 \sqrt {x^{6}-1}}{x^{6}+2}\right )}{6}\) | \(86\) |
risch | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{6}-1}}{x^{3}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}+6 \sqrt {x^{6}-1}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x^{6}+2}\right )}{6}\) | \(90\) |
1/6*(2*arctan(1/3*(x^6-1)^(1/2)*3^(1/2))*3^(1/2)*x^6+3*arctan(1/(x^6-1)^(1 /2))*x^6+(x^6-1)^(1/2))/x^6
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - 3 \, x^{6} \arctan \left (\sqrt {x^{6} - 1}\right ) + \sqrt {x^{6} - 1}}{6 \, x^{6}} \]
1/6*(2*sqrt(3)*x^6*arctan(1/3*sqrt(3)*sqrt(x^6 - 1)) - 3*x^6*arctan(sqrt(x ^6 - 1)) + sqrt(x^6 - 1))/x^6
Time = 8.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{3} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{2} + \frac {\sqrt {x^{6} - 1}}{6 x^{6}} \]
\[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 2\right )} x^{7}} \,d x } \]
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{2} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^6 - 1)) + 1/6*sqrt(x^6 - 1)/x^6 - 1/ 2*arctan(sqrt(x^6 - 1))
Time = 5.93 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx=\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{3}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{2}+\frac {\sqrt {x^6-1}}{6\,x^6} \]