Integrand size = 29, antiderivative size = 41 \[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\frac {4}{3} \arctan \left (\sqrt {-1+x^6}\right )-\frac {\arctan \left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}} \] Output:
4/3*arctan((x^6-1)^(1/2))-1/6*arctan(2/3*(x^6-1)^(1/2)*3^(1/2))*3^(1/2)
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\frac {4}{3} \arctan \left (\sqrt {-1+x^6}\right )-\frac {\arctan \left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}} \] Input:
Integrate[(-4 + 13*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]
Output:
(4*ArcTan[Sqrt[-1 + x^6]])/3 - ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]]/(2*Sqrt[ 3])
Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1043, 174, 73, 216, 217}
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.
\(\displaystyle \int \frac {13 x^6-4}{x \sqrt {x^6-1} \left (4 x^6-1\right )} \, dx\) |
\(\Big \downarrow \) 1043 |
\(\displaystyle \frac {1}{6} \int \frac {4-13 x^6}{x^6 \left (1-4 x^6\right ) \sqrt {x^6-1}}dx^6\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{6} \left (4 \int \frac {1}{x^6 \sqrt {x^6-1}}dx^6+3 \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^6\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} \left (6 \int \frac {1}{-4 x^{12}-3}d\sqrt {x^6-1}+8 \int \frac {1}{x^{12}+1}d\sqrt {x^6-1}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{6} \left (6 \int \frac {1}{-4 x^{12}-3}d\sqrt {x^6-1}+8 \arctan \left (\sqrt {x^6-1}\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{6} \left (8 \arctan \left (\sqrt {x^6-1}\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )\right )\) |
Input:
Int[(-4 + 13*x^6)/(x*Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]
Output:
(8*ArcTan[Sqrt[-1 + x^6]] - Sqrt[3]*ArcTan[(2*Sqrt[-1 + x^6])/Sqrt[3]])/6
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[(((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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[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 = 0.82 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{3}-2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{3}+2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right )}{12}-\frac {4 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(58\) |
trager | \(\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\) | \(86\) |
Input:
int((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x,method=_RETURNVERBOSE)
Output:
-1/12*3^(1/2)*arctan(1/3*(x^3-2)*3^(1/2)/(x^6-1)^(1/2))+1/12*3^(1/2)*arcta n(1/3*(x^3+2)*3^(1/2)/(x^6-1)^(1/2))-4/3*arctan(1/(x^6-1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {4}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \] Input:
integrate((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="fricas")
Output:
-1/6*sqrt(3)*arctan(2/3*sqrt(3)*sqrt(x^6 - 1)) + 4/3*arctan(sqrt(x^6 - 1))
Time = 10.85 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=- \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{6} + \frac {4 \operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{3} \] Input:
integrate((13*x**6-4)/x/(x**6-1)**(1/2)/(4*x**6-1),x)
Output:
-sqrt(3)*atan(2*sqrt(3)*sqrt(x**6 - 1)/3)/6 + 4*atan(sqrt(x**6 - 1))/3
\[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\int { \frac {13 \, x^{6} - 4}{{\left (4 \, x^{6} - 1\right )} \sqrt {x^{6} - 1} x} \,d x } \] Input:
integrate((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="maxima")
Output:
integrate((13*x^6 - 4)/((4*x^6 - 1)*sqrt(x^6 - 1)*x), x)
Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {4}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \] Input:
integrate((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="giac")
Output:
-1/6*sqrt(3)*arctan(2/3*sqrt(3)*sqrt(x^6 - 1)) + 4/3*arctan(sqrt(x^6 - 1))
Time = 7.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\frac {4\,\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{6} \] Input:
int((13*x^6 - 4)/(x*(x^6 - 1)^(1/2)*(4*x^6 - 1)),x)
Output:
(4*atan((x^6 - 1)^(1/2)))/3 - (3^(1/2)*atan((2*3^(1/2)*(x^6 - 1)^(1/2))/3) )/6
\[ \int \frac {-4+13 x^6}{x \sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=-4 \left (\int \frac {\sqrt {x^{6}-1}}{4 x^{13}-5 x^{7}+x}d x \right )+13 \left (\int \frac {\sqrt {x^{6}-1}\, x^{5}}{4 x^{12}-5 x^{6}+1}d x \right ) \] Input:
int((13*x^6-4)/x/(x^6-1)^(1/2)/(4*x^6-1),x)
Output:
- 4*int(sqrt(x**6 - 1)/(4*x**13 - 5*x**7 + x),x) + 13*int((sqrt(x**6 - 1) *x**5)/(4*x**12 - 5*x**6 + 1),x)