Integrand size = 30, antiderivative size = 190 \[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\frac {4 \left (1+2 x^2+x^4\right ) \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2\ 2^{3/4}}-\frac {3 \arctan \left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{4 \sqrt [4]{2}}-\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2\ 2^{3/4}}-\frac {3 \text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{4 \sqrt [4]{2}} \]
4/9*(x^4+2*x^2+1)*(x^5+x^3)^(1/4)/x^3+3/4*2^(1/4)*arctan(2^(1/4)*x/(x^5+x^ 3)^(1/4))-3/8*arctan(2^(3/4)*x*(x^5+x^3)^(1/4)/(2^(1/2)*x^2-(x^5+x^3)^(1/2 )))*2^(3/4)-3/4*2^(1/4)*arctanh(2^(1/4)*x/(x^5+x^3)^(1/4))-3/8*arctanh((1/ 2*x^2*2^(3/4)+1/2*(x^5+x^3)^(1/2)*2^(1/4))/x/(x^5+x^3)^(1/4))*2^(3/4)
Time = 0.95 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\frac {\sqrt [4]{x^3+x^5} \left (32 \sqrt [4]{1+x^2}+64 x^2 \sqrt [4]{1+x^2}+32 x^4 \sqrt [4]{1+x^2}+54 \sqrt [4]{2} x^{9/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-27\ 2^{3/4} x^{9/4} \arctan \left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )-54 \sqrt [4]{2} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-27\ 2^{3/4} x^{9/4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{72 x^3 \sqrt [4]{1+x^2}} \]
((x^3 + x^5)^(1/4)*(32*(1 + x^2)^(1/4) + 64*x^2*(1 + x^2)^(1/4) + 32*x^4*( 1 + x^2)^(1/4) + 54*2^(1/4)*x^(9/4)*ArcTan[(2^(1/4)*x^(1/4))/(1 + x^2)^(1/ 4)] - 27*2^(3/4)*x^(9/4)*ArcTan[(2^(3/4)*x^(1/4)*(1 + x^2)^(1/4))/(Sqrt[2] *Sqrt[x] - Sqrt[1 + x^2])] - 54*2^(1/4)*x^(9/4)*ArcTanh[(2^(1/4)*x^(1/4))/ (1 + x^2)^(1/4)] - 27*2^(3/4)*x^(9/4)*ArcTanh[(2*2^(1/4)*x^(1/4)*(1 + x^2) ^(1/4))/(2*Sqrt[x] + Sqrt[2]*Sqrt[1 + x^2])]))/(72*x^3*(1 + x^2)^(1/4))
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.
| \(\displaystyle \int \frac {\sqrt [4]{x^5+x^3} \left (x^8+x^4+1\right )}{x^4 \left (x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^5+x^3} \int -\frac {\sqrt [4]{x^2+1} \left (x^8+x^4+1\right )}{x^{13/4} \left (1-x^4\right )}dx}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^5+x^3} \int \frac {\sqrt [4]{x^2+1} \left (x^8+x^4+1\right )}{x^{13/4} \left (1-x^4\right )}dx}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{13/4} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}dx}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (-\frac {x^{7/2}}{\left (x^2+1\right )^{3/4}}-\frac {x^{3/2}}{\left (x^2+1\right )^{3/4}}-\frac {3 \sqrt {x}}{2 (x+1) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}-1\right ) \left (x^2+1\right )^{3/4}}-\frac {3}{4 \left (\sqrt {x}+1\right ) \left (x^2+1\right )^{3/4}}+\frac {1}{\left (x^2+1\right )^{3/4} \sqrt {x}}+\frac {1}{\left (x^2+1\right )^{3/4} x^{5/2}}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \frac {x^8+x^4+1}{x^{5/2} \left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\) |
3.24.82.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 26.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.24
| method | result | size |
| pseudoelliptic | \(\frac {-54 x^{3} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right )+\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right )\right ) 2^{\frac {1}{4}}-27 x^{3} \left (\ln \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{-2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )\right ) 2^{\frac {3}{4}}+64 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{2}+1\right )^{2}}{144 x^{3}}\) | \(235\) |
| trager | \(\text {Expression too large to display}\) | \(735\) |
| risch | \(\text {Expression too large to display}\) | \(1774\) |
1/144*(-54*x^3*(2*arctan(1/2*2^(3/4)/x*(x^3*(x^2+1))^(1/4))+ln((-2^(1/4)*x -(x^3*(x^2+1))^(1/4))/(2^(1/4)*x-(x^3*(x^2+1))^(1/4))))*2^(1/4)-27*x^3*(ln ((2^(3/4)*(x^3*(x^2+1))^(1/4)*x+2^(1/2)*x^2+(x^3*(x^2+1))^(1/2))/(-2^(3/4) *(x^3*(x^2+1))^(1/4)*x+2^(1/2)*x^2+(x^3*(x^2+1))^(1/2)))+2*arctan((2^(1/4) *(x^3*(x^2+1))^(1/4)+x)/x)+2*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)-x)/x))*2^ (3/4)+64*(x^3*(x^2+1))^(1/4)*(x^2+1)^2)/x^3
Result contains complex when optimal does not.
Time = 3.68 (sec) , antiderivative size = 763, normalized size of antiderivative = 4.02 \[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\text {Too large to display} \]
1/1152*((27*I + 27)*8^(3/4)*sqrt(2)*x^3*log(((I + 1)*8^(3/4)*sqrt(2)*sqrt( x^5 + x^3)*x - 8*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*(-(I - 1)*x^4 + (2*I - 2)*x^3 - (I - 1)*x^2) - 8*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) - (27*I - 27)*8^(3/4)*sqrt(2)*x^3*log((-(I - 1)*8^(3/4)*sqrt(2)*sq rt(x^5 + x^3)*x + 8*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*((I + 1)*x^4 - (2*I + 2)*x^3 + (I + 1)*x^2) - 8*(x^5 + x^3)^(3/4))/(x^4 + 2*x^ 3 + x^2)) + (27*I - 27)*8^(3/4)*sqrt(2)*x^3*log(((I - 1)*8^(3/4)*sqrt(2)*s qrt(x^5 + x^3)*x + 8*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 8^(1/4)*sqrt(2)*(-( I + 1)*x^4 + (2*I + 2)*x^3 - (I + 1)*x^2) - 8*(x^5 + x^3)^(3/4))/(x^4 + 2* x^3 + x^2)) - (27*I + 27)*8^(3/4)*sqrt(2)*x^3*log((-(I + 1)*8^(3/4)*sqrt(2 )*sqrt(x^5 + x^3)*x - 8*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 8^(1/4)*sqrt(2)* ((I - 1)*x^4 - (2*I - 2)*x^3 + (I - 1)*x^2) - 8*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) - 54*8^(3/4)*x^3*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 8^(3 /4)*sqrt(x^5 + x^3)*x + 8^(1/4)*(x^4 + 2*x^3 + x^2) + 4*(x^5 + x^3)^(3/4)) /(x^4 - 2*x^3 + x^2)) + 54*I*8^(3/4)*x^3*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4) *x^2 + I*8^(3/4)*sqrt(x^5 + x^3)*x - 8^(1/4)*(I*x^4 + 2*I*x^3 + I*x^2) - 4 *(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 54*I*8^(3/4)*x^3*log(-(4*sqrt(2 )*(x^5 + x^3)^(1/4)*x^2 - I*8^(3/4)*sqrt(x^5 + x^3)*x - 8^(1/4)*(-I*x^4 - 2*I*x^3 - I*x^2) - 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) + 54*8^(3/4)* x^3*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 8^(3/4)*sqrt(x^5 + x^3)*x - ...
\[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
Integral((x**3*(x**2 + 1))**(1/4)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x* *2 + 1)/(x**4*(x - 1)*(x + 1)*(x**2 + 1)), x)
\[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\int { \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}} \,d x } \]
\[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\int { \frac {{\left (x^{8} + x^{4} + 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx=\int \frac {{\left (x^5+x^3\right )}^{1/4}\,\left (x^8+x^4+1\right )}{x^4\,\left (x^4-1\right )} \,d x \]