Integrand size = 31, antiderivative size = 351 \[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=\frac {\arctan \left (\frac {3^{7/8} \sqrt {2-\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )}{2^{3/4} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {\sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \arctan \left (\frac {3^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )}{2^{3/4}}+\frac {\text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right )}{2^{3/4} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {\sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2+\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right )}{2^{3/4}} \]
1/2*arctan(3^(7/8)*(2-2^(1/2))^(1/2)*x*(x^5-x^3)^(1/4)/(-3*x^2+3^(3/4)*(x^ 5-x^3)^(1/2)))*2^(1/4)/(51+36*2^(1/2))^(1/8)+1/2*(17/3+4*2^(1/2))^(1/8)*ar ctan(3^(7/8)*(2+2^(1/2))^(1/2)*x*(x^5-x^3)^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^3) ^(1/2)))*2^(1/4)+1/2*arctanh((3^(1/8)*x^2/(2-2^(1/2))^(1/2)+1/3*(x^5-x^3)^ (1/2)*3^(7/8)/(2-2^(1/2))^(1/2))/x/(x^5-x^3)^(1/4))*2^(1/4)/(51+36*2^(1/2) )^(1/8)+1/2*(17/3+4*2^(1/2))^(1/8)*arctanh((3^(1/8)*x^2/(2+2^(1/2))^(1/2)+ 1/3*(x^5-x^3)^(1/2)*3^(7/8)/(2+2^(1/2))^(1/2))/x/(x^5-x^3)^(1/4))*2^(1/4)
Time = 2.75 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.94 \[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=\frac {x^{3/4} \sqrt [4]{-1+x^2} \left (\arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+\sqrt [4]{17+12 \sqrt {2}} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+\sqrt [4]{17+12 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )\right )}{2^{3/4} \sqrt [8]{51+36 \sqrt {2}} \sqrt [4]{x^3 \left (-1+x^2\right )}} \]
(x^(3/4)*(-1 + x^2)^(1/4)*(ArcTan[(2^(1/4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1/4 ))/((17 + 12*Sqrt[2])^(1/8)*(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + (17 + 12*Sqrt[2])^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x^(1/4 )*(-1 + x^2)^(1/4))/(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2])] + ArcTanh[(2^(1 /4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1/4))/((17 + 12*Sqrt[2])^(1/8)*(3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + (17 + 12*Sqrt[2])^(1/4)*ArcTanh[(2^(1/4)*3^( 7/8)*(17 + 12*Sqrt[2])^(1/8)*x^(1/4)*(-1 + x^2)^(1/4))/(3*Sqrt[x] + 3^(3/4 )*Sqrt[-1 + x^2])]))/(2^(3/4)*(51 + 36*Sqrt[2])^(1/8)*(x^3*(-1 + x^2))^(1/ 4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.63 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.56, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2467, 1388, 2035, 25, 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.
| \(\displaystyle \int \frac {1-x^4}{\left (x^4+x^2+1\right ) \sqrt [4]{x^5-x^3}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2-1} \int \frac {1-x^4}{x^{3/4} \sqrt [4]{x^2-1} \left (x^4+x^2+1\right )}dx}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \frac {x^{3/4} \sqrt [4]{x^2-1} \int \frac {\left (-x^2-1\right ) \left (x^2-1\right )^{3/4}}{x^{3/4} \left (x^4+x^2+1\right )}dx}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \int -\frac {\left (x^2-1\right )^{3/4} \left (x^2+1\right )}{x^4+x^2+1}d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^2+1\right )}{x^4+x^2+1}d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {4 x^{3/4} \sqrt [4]{x^2-1} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \left (x^2-1\right )^{3/4}}{2 x^2-i \sqrt {3}+1}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \left (x^2-1\right )^{3/4}}{2 x^2+i \sqrt {3}+1}\right )d\sqrt [4]{x}}{\sqrt [4]{x^5-x^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 x^{3/4} \sqrt [4]{x^2-1} \left (-\frac {\left (\sqrt {3}+3 i\right ) \sqrt [4]{x} \left (x^2-1\right )^{3/4} \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{3 \left (\sqrt {3}+i\right ) \left (1-x^2\right )^{3/4}}-\frac {\left (-\sqrt {3}+3 i\right ) \sqrt [4]{x} \left (x^2-1\right )^{3/4} \operatorname {AppellF1}\left (\frac {1}{8},-\frac {3}{4},1,\frac {9}{8},x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{3 \left (-\sqrt {3}+i\right ) \left (1-x^2\right )^{3/4}}\right )}{\sqrt [4]{x^5-x^3}}\) |
(4*x^(3/4)*(-1 + x^2)^(1/4)*(-1/3*((3*I + Sqrt[3])*x^(1/4)*(-1 + x^2)^(3/4 )*AppellF1[1/8, -3/4, 1, 9/8, x^2, (-2*x^2)/(1 - I*Sqrt[3])])/((I + Sqrt[3 ])*(1 - x^2)^(3/4)) - ((3*I - Sqrt[3])*x^(1/4)*(-1 + x^2)^(3/4)*AppellF1[1 /8, -3/4, 1, 9/8, x^2, (-2*x^2)/(1 + I*Sqrt[3])])/(3*(I - Sqrt[3])*(1 - x^ 2)^(3/4))))/(-x^3 + x^5)^(1/4)
3.30.42.3.1 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_)/((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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 33.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.11
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{2}\) | \(37\) |
| trager | \(\text {Expression too large to display}\) | \(2140\) |
Result contains complex when optimal does not.
Time = 132.86 (sec) , antiderivative size = 2012, normalized size of antiderivative = 5.73 \[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=\text {Too large to display} \]
-(1/24*I + 1/24)*3^(7/8)*sqrt(2)*(-1)^(1/8)*log((3^(7/8)*sqrt(2)*(-1)^(1/8 )*((109*I + 109)*x^6 - (264*I + 264)*x^5 - (545*I + 545)*x^4 + (264*I + 26 4)*x^3 + (109*I + 109)*x^2) - 6*3^(3/8)*sqrt(2)*(-1)^(5/8)*(-(22*I + 22)*x ^6 - (109*I + 109)*x^5 + (110*I + 110)*x^4 + (109*I + 109)*x^3 - (22*I + 2 2)*x^2) - 12*(x^5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(44*I*x^2 + 109*I*x - 44 *I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*sqrt(2)*(-1)^(7/8)*(-(109* I - 109)*x^3 + (132*I - 132)*x^2 + (109*I - 109)*x) + 3^(5/8)*sqrt(2)*(-1) ^(3/8)*((44*I - 44)*x^3 + (109*I - 109)*x^2 - (44*I - 44)*x)) - 12*(x^5 - x^3)^(1/4)*(3^(1/4)*(-1)^(3/4)*(109*I*x^4 - 132*I*x^3 - 109*I*x^2) + 3^(3/ 4)*(-1)^(1/4)*(-44*I*x^4 - 109*I*x^3 + 44*I*x^2)))/(x^6 + x^4 + x^2)) + (1 /24*I - 1/24)*3^(7/8)*sqrt(2)*(-1)^(1/8)*log((3^(7/8)*sqrt(2)*(-1)^(1/8)*( -(109*I - 109)*x^6 + (264*I - 264)*x^5 + (545*I - 545)*x^4 - (264*I - 264) *x^3 - (109*I - 109)*x^2) - 6*3^(3/8)*sqrt(2)*(-1)^(5/8)*((22*I - 22)*x^6 + (109*I - 109)*x^5 - (110*I - 110)*x^4 - (109*I - 109)*x^3 + (22*I - 22)* x^2) - 12*(x^5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(44*I*x^2 + 109*I*x - 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*sqrt(2)*(-1)^(7/8)*((109*I + 109)*x^3 - (132*I + 132)*x^2 - (109*I + 109)*x) + 3^(5/8)*sqrt(2)*(-1)^(3/ 8)*(-(44*I + 44)*x^3 - (109*I + 109)*x^2 + (44*I + 44)*x)) - 12*(x^5 - x^3 )^(1/4)*(3^(3/4)*(-1)^(1/4)*(44*I*x^4 + 109*I*x^3 - 44*I*x^2) + 3^(1/4)*(- 1)^(3/4)*(-109*I*x^4 + 132*I*x^3 + 109*I*x^2)))/(x^6 + x^4 + x^2)) + (1...
\[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=- \int \frac {x^{4}}{x^{4} \sqrt [4]{x^{5} - x^{3}} + x^{2} \sqrt [4]{x^{5} - x^{3}} + \sqrt [4]{x^{5} - x^{3}}}\, dx - \int \left (- \frac {1}{x^{4} \sqrt [4]{x^{5} - x^{3}} + x^{2} \sqrt [4]{x^{5} - x^{3}} + \sqrt [4]{x^{5} - x^{3}}}\right )\, dx \]
-Integral(x**4/(x**4*(x**5 - x**3)**(1/4) + x**2*(x**5 - x**3)**(1/4) + (x **5 - x**3)**(1/4)), x) - Integral(-1/(x**4*(x**5 - x**3)**(1/4) + x**2*(x **5 - x**3)**(1/4) + (x**5 - x**3)**(1/4)), x)
\[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=\int { -\frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}} \,d x } \]
\[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=\int { -\frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx=-\int \frac {x^4-1}{{\left (x^5-x^3\right )}^{1/4}\,\left (x^4+x^2+1\right )} \,d x \]