\(\int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} (-1+4 x^6)} \, dx\) [851]

Optimal result

Integrand size = 30, antiderivative size = 64 \[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {4 x^6}{\sqrt {3}}-\frac {4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {5}{6} \log \left (x^3+\sqrt {-1+x^6}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\frac {1}{6} \left (\sqrt {3} \arctan \left (\frac {1-4 x^6+4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )-5 \log \left (-x^3+\sqrt {-1+x^6}\right )\right ) \] Input:

Integrate[(-x^2 + 10*x^8)/(Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2027, 1045, 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.

\(\displaystyle \int \frac {10 x^8-x^2}{\sqrt {x^6-1} \left (4 x^6-1\right )} \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {x^2 \left (10 x^6-1\right )}{\sqrt {x^6-1} \left (4 x^6-1\right )}dx\)

\(\Big \downarrow \) 1045

\(\displaystyle \frac {1}{3} \int \frac {1-10 x^6}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {1}{3} \left (\frac {5}{2} \int \frac {1}{\sqrt {x^6-1}}dx^3-\frac {3}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{3} \left (\frac {5}{2} \int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}-\frac {3}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {5}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {3}{2} \int \frac {1}{\left (1-4 x^6\right ) \sqrt {x^6-1}}dx^3\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{3} \left (\frac {5}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {3}{2} \int \frac {1}{3 x^6+1}d\frac {x^3}{\sqrt {x^6-1}}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} \left (\frac {5}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )\right )\)

Input:

Int[(-x^2 + 10*x^8)/(Sqrt[-1 + x^6]*(-1 + 4*x^6)),x]
 

Output:

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

Defintions of rubi rules used

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 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]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{3}-2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{3}+2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right )}{12}\)	\(62\)
trager	\(-\frac {5 \ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}+6 x^{3} \sqrt {x^{6}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\)	\(77\)

Input:

int((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {4}{3} \, \sqrt {3} \sqrt {x^{6} - 1} x^{3} - \frac {1}{3} \, \sqrt {3} {\left (4 \, x^{6} - 1\right )}\right ) - \frac {5}{6} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \] Input:

integrate((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\int \frac {x^{2} \cdot \left (10 x^{6} - 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right )}\, dx \] Input:

integrate((10*x**8-x**2)/(x**6-1)**(1/2)/(4*x**6-1),x)
 

Output:

Integral(x**2*(10*x**6 - 1)/(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x 
 + 1))*(2*x**3 - 1)*(2*x**3 + 1)), x)
 

Maxima [F]

\[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\int { \frac {10 \, x^{8} - x^{2}}{{\left (4 \, x^{6} - 1\right )} \sqrt {x^{6} - 1}} \,d x } \] Input:

integrate((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x, algorithm="maxima")
 

Output:

integrate((10*x^8 - x^2)/((4*x^6 - 1)*sqrt(x^6 - 1)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((10*x^8-x^2)/(x^6-1)^(1/2)/(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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=\int -\frac {x^2-10\,x^8}{\sqrt {x^6-1}\,\left (4\,x^6-1\right )} \,d x \] Input:

int(-(x^2 - 10*x^8)/((x^6 - 1)^(1/2)*(4*x^6 - 1)),x)
 

Output:

int(-(x^2 - 10*x^8)/((x^6 - 1)^(1/2)*(4*x^6 - 1)), x)
 

Reduce [F]

\[ \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx=10 \left (\int \frac {\sqrt {x^{6}-1}\, x^{8}}{4 x^{12}-5 x^{6}+1}d x \right )-\left (\int \frac {\sqrt {x^{6}-1}\, x^{2}}{4 x^{12}-5 x^{6}+1}d x \right ) \] Input:

int((10*x^8-x^2)/(x^6-1)^(1/2)/(4*x^6-1),x)
 

Output:

10*int((sqrt(x**6 - 1)*x**8)/(4*x**12 - 5*x**6 + 1),x) - int((sqrt(x**6 - 
1)*x**2)/(4*x**12 - 5*x**6 + 1),x)