Integrand size = 27, antiderivative size = 211 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=-2 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {3}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\frac {x^{9/4} (1+x)^{3/4} \left (16 \left (-\arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )+\text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )\right )-3 \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (1+x)\right )^{3/4}} \] Input:
Integrate[((-1 + x^2)*(x^3 + x^4)^(1/4))/(1 + x^2 + x^4),x]
Output:
(x^(9/4)*(1 + x)^(3/4)*(16*(-ArcTan[(x/(1 + x))^(1/4)] + ArcTanh[(x/(1 + x ))^(1/4)]) - 3*RootSum[3 - 3*#1^4 + #1^8 & , (-Log[x] + 4*Log[(1 + x)^(1/4 ) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(1 + x)^(1/4) - x^(1/4)*#1]*#1^4)/(- 3*#1^3 + 2*#1^7) & ] + RootSum[1 - #1^4 + #1^8 & , (Log[x] - 4*Log[(1 + x) ^(1/4) - x^(1/4)*#1] + Log[x]*#1^4 - 4*Log[(1 + x)^(1/4) - x^(1/4)*#1]*#1^ 4)/(-#1^3 + 2*#1^7) & ]))/(8*(x^3*(1 + x))^(3/4))
Result contains complex when optimal does not.
Time = 7.25 (sec) , antiderivative size = 1944, normalized size of antiderivative = 9.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2467, 25, 2003, 2035, 7279, 2009}
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.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt [4]{x^4+x^3}}{x^4+x^2+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int -\frac {x^{3/4} \sqrt [4]{x+1} \left (1-x^2\right )}{x^4+x^2+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {x^{3/4} \sqrt [4]{x+1} \left (1-x^2\right )}{x^4+x^2+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle -\frac {\sqrt [4]{x^4+x^3} \int \frac {(1-x) x^{3/4} (x+1)^{5/4}}{x^4+x^2+1}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \frac {(1-x) x^{3/2} (x+1)^{5/4}}{x^4+x^2+1}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \int \left (\frac {\left (2-\sqrt [4]{x}\right ) (x+1)^{5/4}}{8 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (\sqrt [4]{x}+2\right ) (x+1)^{5/4}}{8 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}+\frac {\left (\sqrt {x}-2\right ) (x+1)^{5/4}}{4 \left (x-\sqrt {x}+1\right )}+\frac {(1-x) \sqrt {x} (x+1)^{5/4}}{2 \left (x^2-x+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^4+x^3} \left (-\frac {1}{24} \sqrt [4]{x+1} \left (3-\sqrt {x}\right )+\frac {\left (i-\sqrt {3}\right ) x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1-i \sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right )}-\frac {1}{24} \left (1-i \sqrt {3}\right ) x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1-i \sqrt {3}}\right )+\frac {x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,\frac {2 x}{1-i \sqrt {3}}\right )}{3 \sqrt {3} \left (i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1+i \sqrt {3}}\right )}{12 \left (i-\sqrt {3}\right )}-\frac {1}{24} \left (1+i \sqrt {3}\right ) x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},-x,-\frac {2 x}{1+i \sqrt {3}}\right )+\frac {x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},-x,\frac {2 x}{1-i \sqrt {3}}\right )}{3 \sqrt {3} \left (i+\sqrt {3}\right )}-\frac {x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},\frac {2 x}{1+i \sqrt {3}},-x\right )}{3 \sqrt {3} \left (i-\sqrt {3}\right )}-\frac {x^{3/4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x}{1+i \sqrt {3}},-x\right )}{3 \sqrt {3} \left (i-\sqrt {3}\right )}+\frac {9}{16} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )+\frac {\left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{x+1}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right )}+\frac {i \left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{x+1}}\right )}{\sqrt {3}}-\frac {\left (i+\sqrt {3}\right ) \sqrt [4]{-\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{x+1}}\right )}{2\ 3^{3/4}}+\frac {i \sqrt [4]{-\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{x+1}}\right )}{2\ 3^{3/4}}-\frac {9}{16} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {\left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{x+1}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right )}-\frac {i \left (\frac {i-\sqrt {3}}{3 i-\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i-\sqrt {3}}{3 i-\sqrt {3}}} \sqrt [4]{x+1}}\right )}{\sqrt {3}}+\frac {\left (i+\sqrt {3}\right ) \sqrt [4]{-\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{x+1}}\right )}{2\ 3^{3/4}}-\frac {i \sqrt [4]{-\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{\frac {i+\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{x+1}}\right )}{2\ 3^{3/4}}-\frac {\left (1+i \sqrt {3}\right )^2 \left (1+\frac {1}{x}\right )^{3/4} x^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\sqrt {x}\right ),2\right )}{32 (x+1)^{3/4}}-\frac {\left (1-i \sqrt {3}\right ) \left (1+\frac {1}{x}\right )^{3/4} x^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\sqrt {x}\right ),2\right )}{16 (x+1)^{3/4}}-\frac {\left (1+i \sqrt {3}\right )^2 \sqrt {-x} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x+1}\right ),-1\right )}{64 \sqrt {x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt {-x} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x+1}\right ),-1\right )}{32 \sqrt {x}}-\frac {\left (1+i \sqrt {3}\right )^2 \sqrt {-x} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x+1}\right ),-1\right )}{64 \sqrt {x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt {-x} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x+1}\right ),-1\right )}{32 \sqrt {x}}-\frac {1}{64} \left (1+i \sqrt {3}\right )^2 \sqrt {\frac {1}{x+1}} \sqrt {x+1} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}}},\arcsin \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right ),-1\right )-\frac {1}{32} \left (1-i \sqrt {3}\right ) \sqrt {\frac {1}{x+1}} \sqrt {x+1} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}}},\arcsin \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right ),-1\right )-\frac {1}{64} \left (1+i \sqrt {3}\right )^2 \sqrt {\frac {1}{x+1}} \sqrt {x+1} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}}},\arcsin \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right ),-1\right )-\frac {1}{32} \left (1-i \sqrt {3}\right ) \sqrt {\frac {1}{x+1}} \sqrt {x+1} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}}},\arcsin \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right ),-1\right )+\frac {1}{192} \left (1+i \sqrt {3}\right ) \left (\left (1+i \sqrt {3}\right )^2 \sqrt {x}+6 \left (1-i \sqrt {3}\right )\right ) \sqrt [4]{x+1}-\frac {1}{8} x^{3/4} \sqrt [4]{x+1}+\frac {1}{32} \left (1+i \sqrt {3}\right )^2 \sqrt [4]{x} \sqrt [4]{x+1}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt [4]{x} \sqrt [4]{x+1}\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
Input:
Int[((-1 + x^2)*(x^3 + x^4)^(1/4))/(1 + x^2 + x^4),x]
Output:
(-4*(x^3 + x^4)^(1/4)*(-1/24*((3 - Sqrt[x])*(1 + x)^(1/4)) + ((1 + I*Sqrt[ 3])*(6*(1 - I*Sqrt[3]) + (1 + I*Sqrt[3])^2*Sqrt[x])*(1 + x)^(1/4))/192 + ( (1 - I*Sqrt[3])*x^(1/4)*(1 + x)^(1/4))/16 + ((1 + I*Sqrt[3])^2*x^(1/4)*(1 + x)^(1/4))/32 - (x^(3/4)*(1 + x)^(1/4))/8 - ((1 - I*Sqrt[3])*x^(3/4)*Appe llF1[3/4, -5/4, 1, 7/4, -x, (-2*x)/(1 - I*Sqrt[3])])/24 + ((I - Sqrt[3])*x ^(3/4)*AppellF1[3/4, -5/4, 1, 7/4, -x, (-2*x)/(1 - I*Sqrt[3])])/(12*(I + S qrt[3])) + (x^(3/4)*AppellF1[3/4, -5/4, 1, 7/4, -x, (2*x)/(1 - I*Sqrt[3])] )/(3*Sqrt[3]*(I + Sqrt[3])) - ((1 + I*Sqrt[3])*x^(3/4)*AppellF1[3/4, -5/4, 1, 7/4, -x, (-2*x)/(1 + I*Sqrt[3])])/24 + ((I + Sqrt[3])*x^(3/4)*AppellF1 [3/4, -5/4, 1, 7/4, -x, (-2*x)/(1 + I*Sqrt[3])])/(12*(I - Sqrt[3])) + (x^( 3/4)*AppellF1[3/4, -1/4, 1, 7/4, -x, (2*x)/(1 - I*Sqrt[3])])/(3*Sqrt[3]*(I + Sqrt[3])) - (x^(3/4)*AppellF1[3/4, 1, -5/4, 7/4, (2*x)/(1 + I*Sqrt[3]), -x])/(3*Sqrt[3]*(I - Sqrt[3])) - (x^(3/4)*AppellF1[3/4, 1, -1/4, 7/4, (2* x)/(1 + I*Sqrt[3]), -x])/(3*Sqrt[3]*(I - Sqrt[3])) + (9*ArcTan[x^(1/4)/(1 + x)^(1/4)])/16 + (I*((I - Sqrt[3])/(3*I - Sqrt[3]))^(3/4)*ArcTan[x^(1/4)/ (((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/4)*(1 + x)^(1/4))])/Sqrt[3] + (((I - S qrt[3])/(3*I - Sqrt[3]))^(3/4)*ArcTan[x^(1/4)/(((I - Sqrt[3])/(3*I - Sqrt[ 3]))^(1/4)*(1 + x)^(1/4))])/(Sqrt[3]*(I - Sqrt[3])) + ((I/2)*(-((I + Sqrt[ 3])/(3*I + Sqrt[3])))^(1/4)*ArcTan[x^(1/4)/(((I + Sqrt[3])/(3*I + Sqrt[3]) )^(1/4)*(1 + x)^(1/4))])/3^(3/4) - ((I + Sqrt[3])*(-((I + Sqrt[3])/(3*I...
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 76.96 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.76
| method | result | size |
| pseudoelliptic | \(\ln \left (\frac {\left (\left (1+x \right ) x^{3}\right )^{\frac {1}{4}}+x}{x}\right )+2 \arctan \left (\frac {\left (\left (1+x \right ) x^{3}\right )^{\frac {1}{4}}}{x}\right )+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (1+x \right ) x^{3}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-3\right )}\right )}{2}-\ln \left (\frac {-x +\left (\left (1+x \right ) x^{3}\right )^{\frac {1}{4}}}{x}\right )-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (1+x \right ) x^{3}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{2}\) | \(160\) |
| trager | \(\text {Expression too large to display}\) | \(4729\) |
Input:
int((x^2-1)*(x^4+x^3)^(1/4)/(x^4+x^2+1),x,method=_RETURNVERBOSE)
Output:
ln((((1+x)*x^3)^(1/4)+x)/x)+2*arctan(((1+x)*x^3)^(1/4)/x)+3/2*sum((_R^4-1) *ln((-_R*x+((1+x)*x^3)^(1/4))/x)/_R^3/(2*_R^4-3),_R=RootOf(_Z^8-3*_Z^4+3)) -ln((-x+((1+x)*x^3)^(1/4))/x)-1/2*sum((_R^4+1)*ln((-_R*x+((1+x)*x^3)^(1/4) )/x)/_R^3/(2*_R^4-1),_R=RootOf(_Z^8-_Z^4+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.11 (sec) , antiderivative size = 868, normalized size of antiderivative = 4.11 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\text {Too large to display} \] Input:
integrate((x^2-1)*(x^4+x^3)^(1/4)/(x^4+x^2+1),x, algorithm="fricas")
Output:
-1/2*sqrt(-sqrt(1/2*sqrt(-3) + 1/2))*log(((sqrt(-3)*x + x)*sqrt(-sqrt(1/2* sqrt(-3) + 1/2)) + 2*(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(-sqrt(1/2*sqrt(-3) + 1/2))*log(-((sqrt(-3)*x + x)*sqrt(-sqrt(1/2*sqrt(-3) + 1/2)) - 2*(x^4 + x ^3)^(1/4))/x) - 1/2*sqrt(-sqrt(1/2*sqrt(-3) - 3/2))*log(((sqrt(-3)*x + x)* sqrt(-sqrt(1/2*sqrt(-3) - 3/2)) + 2*(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(-sqrt (1/2*sqrt(-3) - 3/2))*log(-((sqrt(-3)*x + x)*sqrt(-sqrt(1/2*sqrt(-3) - 3/2 )) - 2*(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(-sqrt(-1/2*sqrt(-3) + 1/2))*log((( sqrt(-3)*x - x)*sqrt(-sqrt(-1/2*sqrt(-3) + 1/2)) + 2*(x^4 + x^3)^(1/4))/x) - 1/2*sqrt(-sqrt(-1/2*sqrt(-3) + 1/2))*log(-((sqrt(-3)*x - x)*sqrt(-sqrt( -1/2*sqrt(-3) + 1/2)) - 2*(x^4 + x^3)^(1/4))/x) + 1/2*sqrt(-sqrt(-1/2*sqrt (-3) - 3/2))*log(((sqrt(-3)*x - x)*sqrt(-sqrt(-1/2*sqrt(-3) - 3/2)) + 2*(x ^4 + x^3)^(1/4))/x) - 1/2*sqrt(-sqrt(-1/2*sqrt(-3) - 3/2))*log(-((sqrt(-3) *x - x)*sqrt(-sqrt(-1/2*sqrt(-3) - 3/2)) - 2*(x^4 + x^3)^(1/4))/x) - 1/2*( 1/2*sqrt(-3) + 1/2)^(1/4)*log(((sqrt(-3)*x + x)*(1/2*sqrt(-3) + 1/2)^(1/4) + 2*(x^4 + x^3)^(1/4))/x) + 1/2*(1/2*sqrt(-3) + 1/2)^(1/4)*log(-((sqrt(-3 )*x + x)*(1/2*sqrt(-3) + 1/2)^(1/4) - 2*(x^4 + x^3)^(1/4))/x) - 1/2*(1/2*s qrt(-3) - 3/2)^(1/4)*log(((sqrt(-3)*x + x)*(1/2*sqrt(-3) - 3/2)^(1/4) + 2* (x^4 + x^3)^(1/4))/x) + 1/2*(1/2*sqrt(-3) - 3/2)^(1/4)*log(-((sqrt(-3)*x + x)*(1/2*sqrt(-3) - 3/2)^(1/4) - 2*(x^4 + x^3)^(1/4))/x) + 1/2*(-1/2*sqrt( -3) + 1/2)^(1/4)*log(((sqrt(-3)*x - x)*(-1/2*sqrt(-3) + 1/2)^(1/4) + 2*...
Not integrable
Time = 1.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \] Input:
integrate((x**2-1)*(x**4+x**3)**(1/4)/(x**4+x**2+1),x)
Output:
Integral((x**3*(x + 1))**(1/4)*(x - 1)*(x + 1)/((x**2 - x + 1)*(x**2 + x + 1)), x)
Not integrable
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{4} + x^{2} + 1} \,d x } \] Input:
integrate((x^2-1)*(x^4+x^3)^(1/4)/(x^4+x^2+1),x, algorithm="maxima")
Output:
integrate((x^4 + x^3)^(1/4)*(x^2 - 1)/(x^4 + x^2 + 1), x)
Exception generated. \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((x^2-1)*(x^4+x^3)^(1/4)/(x^4+x^2+1),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:proot error [1,0,0,0,1,0,0,0,1]proo t error [1,0,0,0,-1,0,0,0,1]proot error [1,0,-10,0,1]proot error [1,0,-10, 0,1]proot
Not integrable
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x^2-1\right )}{x^4+x^2+1} \,d x \] Input:
int(((x^3 + x^4)^(1/4)*(x^2 - 1))/(x^2 + x^4 + 1),x)
Output:
int(((x^3 + x^4)^(1/4)*(x^2 - 1))/(x^2 + x^4 + 1), x)
Not integrable
Time = 3.37 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.21 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx=\int \frac {x^{\frac {11}{4}} \left (x +1\right )^{\frac {1}{4}}}{x^{4}+x^{2}+1}d x -\left (\int \frac {x^{\frac {3}{4}} \left (x +1\right )^{\frac {1}{4}}}{x^{4}+x^{2}+1}d x \right ) \] Input:
int((x^2-1)*(x^4+x^3)^(1/4)/(x^4+x^2+1),x)
Output:
int((x**(3/4)*(x + 1)**(1/4)*x**2)/(x**4 + x**2 + 1),x) - int((x**(3/4)*(x + 1)**(1/4))/(x**4 + x**2 + 1),x)