\(\int \frac {-2+5 x^6}{x \sqrt {-1+x^6} (2+x^6)} \, dx\) [498]

Optimal result

Integrand size = 27, antiderivative size = 38 \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right )+\frac {2 \arctan \left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

-1/3*arctan((x^6-1)^(1/2))+2/3*arctan(1/3*(x^6-1)^(1/2)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {587, 162, 65, 209} \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \arctan \left (\sqrt {x^6-1}\right ) \]

[In]

Int[(-2 + 5*x^6)/(x*Sqrt[-1 + x^6]*(2 + x^6)),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(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]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 587

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {-2+5 x}{\sqrt {-1+x} x (2+x)} \, dx,x,x^6\right ) \\ & = -\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\right )+\text {Subst}\left (\int \frac {1}{\sqrt {-1+x} (2+x)} \, dx,x,x^6\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\right )+2 \text {Subst}\left (\int \frac {1}{3+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = -\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right )+\frac {2 \arctan \left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right )+\frac {2 \arctan \left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Integrate[(-2 + 5*x^6)/(x*Sqrt[-1 + x^6]*(2 + x^6)),x]

[Out]

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

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}+\frac {2 \arctan \left (\frac {\sqrt {x^{6}-1}\, \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(30\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{6}-1}}{x^{3}}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )-6 \sqrt {x^{6}-1}}{x^{6}+2}\right )}{3}\)	\(78\)

[In]

int((5*x^6-2)/x/(x^6-1)^(1/2)/(x^6+2),x,method=_RETURNVERBOSE)

[Out]

1/3*arctan(1/(x^6-1)^(1/2))+2/3*arctan(1/3*(x^6-1)^(1/2)*3^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

integrate((5*x^6-2)/x/(x^6-1)^(1/2)/(x^6+2),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^6 - 1)) - 1/3*arctan(sqrt(x^6 - 1))

Sympy [A] (verification not implemented)

Time = 9.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{3} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{3} \]

[In]

integrate((5*x**6-2)/x/(x**6-1)**(1/2)/(x**6+2),x)

[Out]

2*sqrt(3)*atan(sqrt(3)*sqrt(x**6 - 1)/3)/3 - atan(sqrt(x**6 - 1))/3

Maxima [F]

\[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\int { \frac {5 \, x^{6} - 2}{{\left (x^{6} + 2\right )} \sqrt {x^{6} - 1} x} \,d x } \]

[In]

integrate((5*x^6-2)/x/(x^6-1)^(1/2)/(x^6+2),x, algorithm="maxima")

[Out]

integrate((5*x^6 - 2)/((x^6 + 2)*sqrt(x^6 - 1)*x), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

integrate((5*x^6-2)/x/(x^6-1)^(1/2)/(x^6+2),x, algorithm="giac")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^6 - 1)) - 1/3*arctan(sqrt(x^6 - 1))

Mupad [B] (verification not implemented)

Time = 5.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {-2+5 x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx=\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{3}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \]

[In]

int((5*x^6 - 2)/(x*(x^6 - 1)^(1/2)*(x^6 + 2)),x)

[Out]

(2*3^(1/2)*atan((3^(1/2)*(x^6 - 1)^(1/2))/3))/3 - atan((x^6 - 1)^(1/2))/3