Integrand size = 25, antiderivative size = 113 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=-\frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \]
Time = 0.36 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.31 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{1+x^2} \left (-\text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{-3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ]\right )}{4 \sqrt [4]{x^2+x^4}} \]
(Sqrt[x]*(1 + x^2)^(1/4)*(-RootSum[3 - 3*#1^4 + #1^8 & , (-Log[Sqrt[x]] + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1])/(-3*#1 + 2*#1^5) & ] + RootSum[1 - #1^4 + #1^8 & , (-Log[Sqrt[x]] + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1])/(-#1 + 2*# 1^5) & ]))/(4*(x^2 + x^4)^(1/4))
Result contains complex when optimal does not.
Time = 2.13 (sec) , antiderivative size = 884, normalized size of antiderivative = 7.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2467, 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.
\(\displaystyle \int \frac {x^4}{\sqrt [4]{x^4+x^2} \left (x^8+x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^{7/2}}{\sqrt [4]{x^2+1} \left (x^8+x^4+1\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {x^4}{\sqrt [4]{x^2+1} \left (x^8+x^4+1\right )}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \left (\frac {1+\frac {i}{\sqrt {3}}}{\left (2 x^4-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}+\frac {1-\frac {i}{\sqrt {3}}}{\left (2 x^4+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}\right )d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (-\frac {\left (3-i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3-i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}}}-\frac {\left (3-i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3-i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}}}-\frac {\left (3+i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{6 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}}}\right )}{\sqrt [4]{x^4+x^2}}\) |
(2*Sqrt[x]*(1 + x^2)^(1/4)*(-1/6*((3 - I*Sqrt[3])*ArcTan[((-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4) )])/((2*(-1 - I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)) - ( (3 - I*Sqrt[3])*ArcTan[((2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2* (-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 - I*Sqrt[3]))^(7/8)*( 2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)) - ((3 + I*Sqrt[3])*ArcTan[((-2 + Sqrt [-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2) ^(1/4))])/(6*(2*(-1 + I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1 /4)) - ((3 + I*Sqrt[3])*ArcTan[((2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[ x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 + I*Sqrt[3])) ^(7/8)*(2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)) - ((3 - I*Sqrt[3])*ArcTanh[(( -2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)* (1 + x^2)^(1/4))])/(6*(2*(-1 - I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 - (2*I)*Sqr t[3]])^(1/4)) - ((3 - I*Sqrt[3])*ArcTanh[((2 + Sqrt[-2 - (2*I)*Sqrt[3]])^( 1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 - I *Sqrt[3]))^(7/8)*(2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)) - ((3 + I*Sqrt[3])* ArcTanh[((-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3 ]))^(1/8)*(1 + x^2)^(1/4))])/(6*(2*(-1 + I*Sqrt[3]))^(7/8)*(-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)) - ((3 + I*Sqrt[3])*ArcTanh[((2 + Sqrt[-2 + (2*I)*S qrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/...
3.17.92.3.1 Defintions of rubi rules used
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 = 102.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{5}-3 \textit {\_R}}\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{5}-\textit {\_R}}\right )}{4}\) | \(100\) |
trager | \(\text {Expression too large to display}\) | \(5897\) |
-1/4*sum(ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(2*_R^5-3*_R),_R=RootOf(_Z^8-3* _Z^4+3))+1/4*sum(ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/(2*_R^5-_R),_R=RootOf(_ Z^8-_Z^4+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 10.15 (sec) , antiderivative size = 3429, normalized size of antiderivative = 30.35 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\text {Too large to display} \]
1/48*sqrt(2)*sqrt(sqrt(6)*sqrt(I*sqrt(3) + 3))*log(-(2*sqrt(6)*(9*x^4 - 24 *x^2 + sqrt(3)*(13*I*x^4 - 2*I*x^2))*(x^4 + x^2)^(1/4)*sqrt(I*sqrt(3) + 3) + 12*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(-8*I*x^2 + 5*I) + 11) - (sqrt(6) *sqrt(x^4 + x^2)*(sqrt(3)*sqrt(2)*(3*I*x^3 - 8*I*x) - sqrt(2)*(13*x^3 - 2* x))*sqrt(I*sqrt(3) + 3) + sqrt(3)*sqrt(2)*(-7*I*x^5 + 37*I*x^3 + 11*I*x) + 3*sqrt(2)*(21*x^5 + x^3 - 5*x))*sqrt(sqrt(6)*sqrt(I*sqrt(3) + 3)))/(x^5 - x^3 + x)) - 1/48*sqrt(2)*sqrt(sqrt(6)*sqrt(I*sqrt(3) + 3))*log(-(2*sqrt(6 )*(9*x^4 - 24*x^2 + sqrt(3)*(13*I*x^4 - 2*I*x^2))*(x^4 + x^2)^(1/4)*sqrt(I *sqrt(3) + 3) + 12*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(-8*I*x^2 + 5*I) + 1 1) - (sqrt(6)*sqrt(x^4 + x^2)*(sqrt(3)*sqrt(2)*(-3*I*x^3 + 8*I*x) + sqrt(2 )*(13*x^3 - 2*x))*sqrt(I*sqrt(3) + 3) + sqrt(3)*sqrt(2)*(7*I*x^5 - 37*I*x^ 3 - 11*I*x) - 3*sqrt(2)*(21*x^5 + x^3 - 5*x))*sqrt(sqrt(6)*sqrt(I*sqrt(3) + 3)))/(x^5 - x^3 + x)) - 1/48*sqrt(2)*sqrt(-sqrt(6)*sqrt(I*sqrt(3) + 3))* log((2*sqrt(6)*(9*x^4 - 24*x^2 - sqrt(3)*(-13*I*x^4 + 2*I*x^2))*(x^4 + x^2 )^(1/4)*sqrt(I*sqrt(3) + 3) - 12*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(-8*I* x^2 + 5*I) + 11) + (sqrt(6)*sqrt(x^4 + x^2)*(sqrt(3)*sqrt(2)*(3*I*x^3 - 8* I*x) - sqrt(2)*(13*x^3 - 2*x))*sqrt(I*sqrt(3) + 3) + sqrt(3)*sqrt(2)*(7*I* x^5 - 37*I*x^3 - 11*I*x) - 3*sqrt(2)*(21*x^5 + x^3 - 5*x))*sqrt(-sqrt(6)*s qrt(I*sqrt(3) + 3)))/(x^5 - x^3 + x)) + 1/48*sqrt(2)*sqrt(-sqrt(6)*sqrt(I* sqrt(3) + 3))*log((2*sqrt(6)*(9*x^4 - 24*x^2 - sqrt(3)*(-13*I*x^4 + 2*I...
Not integrable
Time = 1.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Integral(x**4/((x**2*(x**2 + 1))**(1/4)*(x**2 - x + 1)*(x**2 + x + 1)*(x** 4 - x**2 + 1)), x)
Not integrable
Time = 0.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (x^{8} + x^{4} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
2/21*(4*x^5 + x^3 - 3*x)*x^(7/2)/((x^8 + x^4 + 1)*(x^2 + 1)^(1/4)) - integ rate(8/21*((4*x^4 + x^2 - 3)*x^(15/2) + 2*(4*x^4 + x^2 - 3)*x^(7/2))/((x^1 6 + 2*x^12 + 3*x^8 + 2*x^4 + 1)*(x^2 + 1)^(1/4)), x)
Exception generated. \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\text {Exception raised: TypeError} \]
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 = 5.58 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.22 \[ \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx=\int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^8+x^4+1\right )} \,d x \]