\(\int \frac {x^5 (1-x^4)}{1-x^4+x^8} \, dx\) [63]

Optimal result

Integrand size = 23, antiderivative size = 66 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {x^2}{2}-\frac {1}{4} \arctan \left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \arctan \left (\sqrt {3}+2 x^2\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} x^2}{1+x^4}\right )}{4 \sqrt {3}} \] Output:

-1/2*x^2+1/4*arctan(2*x^2-3^(1/2))+1/4*arctan(3^(1/2)+2*x^2)+1/12*arctanh( 
3^(1/2)*x^2/(x^4+1))*3^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.29 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=\frac {1}{12} i \left (6 i x^2+\sqrt {-6-6 i \sqrt {3}} \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )-\sqrt {-6+6 i \sqrt {3}} \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )\right ) \] Input:

Integrate[(x^5*(1 - x^4))/(1 - x^4 + x^8),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1814, 1602, 25, 1407, 1142, 25, 1083, 217, 1103}

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.

\(\displaystyle \int \frac {x^5 \left (1-x^4\right )}{x^8-x^4+1} \, dx\)

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1602

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1407

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

Int[(x^5*(1 - x^4))/(1 - x^4 + x^8),x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 217

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])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1407

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ 
c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   Int[(r - x)/(q - r* 
x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(r + x)/(q + r*x + x^2), x], x]]] 
 /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
 

rule 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

rule 1814

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e 
_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Sub 
st[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k)))^ 
p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 
 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58

method	result	size
risch	\(-\frac {x^{2}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-3 \textit {\_R}^{3}+x^{2}+\textit {\_R} \right )\right )}{4}\)	\(38\)
default	\(-\frac {x^{2}}{2}+\frac {\sqrt {3}\, \ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{24}+\frac {\arctan \left (2 x^{2}+\sqrt {3}\right )}{4}-\frac {\sqrt {3}\, \ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{24}+\frac {\arctan \left (2 x^{2}-\sqrt {3}\right )}{4}\)	\(70\)

Input:

int(x^5*(-x^4+1)/(x^8-x^4+1),x,method=_RETURNVERBOSE)
 

Output:

-1/2*x^2+1/4*sum(_R*ln(-3*_R^3+x^2+_R),_R=RootOf(9*_Z^4+3*_Z^2+1))
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {1}{2} \, x^{2} + \frac {1}{24} \, \sqrt {3} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) + \frac {1}{4} \, \arctan \left (2 \, x^{2} + \sqrt {3}\right ) - \frac {1}{4} \, \arctan \left (-2 \, x^{2} + \sqrt {3}\right ) \] Input:

integrate(x^5*(-x^4+1)/(x^8-x^4+1),x, algorithm="fricas")
 

Output:

-1/2*x^2 + 1/24*sqrt(3)*log(x^4 + sqrt(3)*x^2 + 1) - 1/24*sqrt(3)*log(x^4 
- sqrt(3)*x^2 + 1) + 1/4*arctan(2*x^2 + sqrt(3)) - 1/4*arctan(-2*x^2 + sqr 
t(3))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=- \frac {x^{2}}{2} - \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\operatorname {atan}{\left (2 x^{2} - \sqrt {3} \right )}}{4} + \frac {\operatorname {atan}{\left (2 x^{2} + \sqrt {3} \right )}}{4} \] Input:

integrate(x**5*(-x**4+1)/(x**8-x**4+1),x)
 

Output:

-x**2/2 - sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/24 + sqrt(3)*log(x**4 + sqr 
t(3)*x**2 + 1)/24 + atan(2*x**2 - sqrt(3))/4 + atan(2*x**2 + sqrt(3))/4
 

Maxima [F]

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

integrate(x^5*(-x^4+1)/(x^8-x^4+1),x, algorithm="maxima")
 

Output:

-1/2*x^2 + integrate(x/(x^8 - x^4 + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {1}{2} \, x^{2} + \frac {1}{24} \, \sqrt {3} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) + \frac {1}{4} \, \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + \frac {1}{4} \, \arctan \left (2 \, x^{2} - \sqrt {3}\right ) \] Input:

integrate(x^5*(-x^4+1)/(x^8-x^4+1),x, algorithm="giac")
 

Output:

-1/2*x^2 + 1/24*sqrt(3)*log(x^4 + sqrt(3)*x^2 + 1) - 1/24*sqrt(3)*log(x^4 
- sqrt(3)*x^2 + 1) + 1/4*arctan(2*x^2 + sqrt(3)) + 1/4*arctan(2*x^2 - sqrt 
(3))
 

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\mathrm {atan}\left (-\frac {x^2}{2}+\frac {\sqrt {3}\,x^2\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {x^2}{2}+\frac {\sqrt {3}\,x^2\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {x^2}{2} \] Input:

int(-(x^5*(x^4 - 1))/(x^8 - x^4 + 1),x)
 

Output:

- atan((3^(1/2)*x^2*1i)/2 - x^2/2)*((3^(1/2)*1i)/12 + 1/4) - atan((3^(1/2) 
*x^2*1i)/2 + x^2/2)*((3^(1/2)*1i)/12 - 1/4) - x^2/2
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 341, normalized size of antiderivative = 5.17 \[ \int \frac {x^5 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}+\frac {\sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}+\frac {\sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}-\frac {\sqrt {3}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{24}-\frac {\sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{24}-\frac {x^{2}}{2} \] Input:

int(x^5*(-x^4+1)/(x^8-x^4+1),x)
 

Output:

( - 3*sqrt( - sqrt(3) + 2)*sqrt(6)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( 
 - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((sqrt(6) + sqrt(2) 
 - 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(6)*atan((s 
qrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2) 
*sqrt(2)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt 
( - sqrt(3) + 2)*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sq 
rt(2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4* 
x)/(sqrt(6) + sqrt(2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(6)*atan((2*sqrt( - s 
qrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*a 
tan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) + sqrt(3)*log( - s 
qrt( - sqrt(3) + 2)*x + x**2 + 1) + sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x 
**2 + 1) - sqrt(3)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2) - sqrt(3 
)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2) - 12*x**2)/24