3.12.45 \(\int \frac {\sqrt {-1+x^6} (-1+2 x^6)^2}{x^4 (-1+4 x^6)} \, dx\) [1145]

3.12.45.1 Optimal result

Integrand size = 31, antiderivative size = 85 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^4 \left (-1+4 x^6\right )} \, dx=\frac {\sqrt {-1+x^6} \left (2+x^6\right )}{6 x^3}+\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {4 x^6}{\sqrt {3}}-\frac {4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {5}{12} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

output

1/6*(x^6-1)^(1/2)*(x^6+2)/x^3-1/12*arctan(-1/3*3^(1/2)+4/3*x^6*3^(1/2)+4/3 
*x^3*(x^6-1)^(1/2)*3^(1/2))*3^(1/2)-5/12*ln(x^3+(x^6-1)^(1/2))
 

3.12.45.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^4 \left (-1+4 x^6\right )} \, dx=\frac {1}{12} \left (\frac {2 \sqrt {-1+x^6} \left (2+x^6\right )}{x^3}-\sqrt {3} \arctan \left (\frac {1-4 x^6+4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )+5 \log \left (-x^3+\sqrt {-1+x^6}\right )\right ) \]

input

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

output

((2*Sqrt[-1 + x^6]*(2 + x^6))/x^3 - Sqrt[3]*ArcTan[(1 - 4*x^6 + 4*x^3*Sqrt 
[-1 + x^6])/Sqrt[3]] + 5*Log[-x^3 + Sqrt[-1 + x^6]])/12
 

3.12.45.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.58, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {1045, 25, 448, 403, 27, 398, 224, 219, 291, 216, 442, 25, 398, 224, 219, 291, 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.

\(\displaystyle \int \frac {\sqrt {x^6-1} \left (2 x^6-1\right )^2}{x^4 \left (4 x^6-1\right )} \, dx\)

\(\Big \downarrow \) 1045

\(\displaystyle \frac {1}{3} \int -\frac {\left (1-2 x^6\right )^2 \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{3} \int \frac {\left (1-2 x^6\right )^2 \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\)

\(\Big \downarrow \) 448

\(\displaystyle \frac {1}{3} \left (2 \int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{1-4 x^6}dx^3-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {1}{8} \int \frac {6 \left (1-2 x^6\right )}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \int \frac {1-2 x^6}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {x^6-1}}dx^3+\frac {1}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}+\frac {1}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{3 x^6+1}d\frac {x^3}{\sqrt {x^6-1}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )-\int \frac {\left (1-2 x^6\right ) \sqrt {x^6-1}}{x^6 \left (1-4 x^6\right )}dx^3\right )\)

\(\Big \downarrow \) 442

\(\displaystyle \frac {1}{3} \left (-\int -\frac {2 x^6+1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3+2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\int \frac {2 x^6+1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3+2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {1}{3} \left (-\frac {1}{2} \int \frac {1}{\sqrt {x^6-1}}dx^3+\frac {3}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3+2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{3} \left (-\frac {1}{2} \int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}+\frac {3}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3+2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3+2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )-\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{3 x^6+1}d\frac {x^3}{\sqrt {x^6-1}}+2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )-\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} \left (2 \left (\frac {1}{4} x^3 \sqrt {x^6-1}-\frac {3}{4} \left (\frac {\arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}}+\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right )\right )+\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1}}{x^3}\right )\)

input

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

output

(Sqrt[-1 + x^6]/x^3 + (Sqrt[3]*ArcTan[(Sqrt[3]*x^3)/Sqrt[-1 + x^6]])/2 + 2 
*((x^3*Sqrt[-1 + x^6])/4 - (3*(ArcTan[(Sqrt[3]*x^3)/Sqrt[-1 + x^6]]/(2*Sqr 
t[3]) + ArcTanh[x^3/Sqrt[-1 + x^6]]/2))/4) - ArcTanh[x^3/Sqrt[-1 + x^6]]/2 
)/3
 

3.12.45.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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])
 

rule 219

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])
 

rule 224

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 291

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]
 

rule 398

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]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 442

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1))   Int[(g*x) 
^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 
*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] 
&&  !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
 

rule 448

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e   Int[(g*x)^m*(a + b*x 
^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2   Int[(g*x)^(m 
 + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, 
b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
 

rule 1045

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]
 

3.12.45.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (x^{6}+2\right )}{6 x^{3}}+\frac {5 \ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{12}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}+6 x^{3} \sqrt {x^{6}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{24}\)	\(94\)
risch	\(\frac {x^{12}+x^{6}-2}{6 x^{3} \sqrt {x^{6}-1}}+\frac {5 \ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{12}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}+6 x^{3} \sqrt {x^{6}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{24}\)	\(97\)
pseudoelliptic	\(\frac {4 \sqrt {x^{6}-1}\, x^{6}-\sqrt {3}\, \arctan \left (\frac {\left (x^{3}-2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right ) x^{3}-\sqrt {3}\, \arctan \left (\frac {\left (x^{3}+2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right ) x^{3}-10 \ln \left (x^{3}+\sqrt {x^{6}-1}\right ) x^{3}+8 \sqrt {x^{6}-1}}{24 x^{3}}\)	\(97\)

input

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

output

1/6*(x^6-1)^(1/2)*(x^6+2)/x^3+5/12*ln(-x^3+(x^6-1)^(1/2))-1/24*RootOf(_Z^2 
+3)*ln(-(2*RootOf(_Z^2+3)*x^6+6*x^3*(x^6-1)^(1/2)+RootOf(_Z^2+3))/(2*x^3-1 
)/(2*x^3+1))
 

3.12.45.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^4 \left (-1+4 x^6\right )} \, dx=-\frac {\sqrt {3} x^{3} \arctan \left (\frac {4}{3} \, \sqrt {3} \sqrt {x^{6} - 1} x^{3} - \frac {1}{3} \, \sqrt {3} {\left (4 \, x^{6} - 1\right )}\right ) - 5 \, x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 4 \, x^{3} - 2 \, {\left (x^{6} + 2\right )} \sqrt {x^{6} - 1}}{12 \, x^{3}} \]

input

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

output

-1/12*(sqrt(3)*x^3*arctan(4/3*sqrt(3)*sqrt(x^6 - 1)*x^3 - 1/3*sqrt(3)*(4*x 
^6 - 1)) - 5*x^3*log(-x^3 + sqrt(x^6 - 1)) - 4*x^3 - 2*(x^6 + 2)*sqrt(x^6 
- 1))/x^3
 

3.12.45.6 Sympy [F]

\[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^4 \left (-1+4 x^6\right )} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{6} - 1\right )^{2}}{x^{4} \cdot \left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right )}\, dx \]

input

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

output

Integral(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(2*x**6 - 1)* 
*2/(x**4*(2*x**3 - 1)*(2*x**3 + 1)), x)
 

3.12.45.7 Maxima [F]

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

input

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

output

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

3.12.45.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^4 \left (-1+4 x^6\right )} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:rootof minimal polynomial must be u 
nitary Error: Bad Argument Valuerootof minimal polynomial must be unitary 
Error: Ba
 

3.12.45.9 Mupad [F(-1)]

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

input

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

output

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