Integrand size = 20, antiderivative size = 203 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=-2 \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )+\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt {2}} \]
-2*arctan((1+x)^(1/4)/(1-x)^(1/4))-2*arctanh((1+x)^(1/4)/(1-x)^(1/4))-1/2* ln(1-(1-x)^(1/4)*2^(1/2)/(1+x)^(1/4)+(1-x)^(1/2)/(1+x)^(1/2))*2^(1/2)+1/2* ln(1+(1-x)^(1/4)*2^(1/2)/(1+x)^(1/4)+(1-x)^(1/2)/(1+x)^(1/2))*2^(1/2)-arct an(-1+(1-x)^(1/4)*2^(1/2)/(1+x)^(1/4))*2^(1/2)-arctan(1+(1-x)^(1/4)*2^(1/2 )/(1+x)^(1/4))*2^(1/2)
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=-2 \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x^2}}{\sqrt {1-x}-\sqrt {1+x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-x^2}}{\sqrt {1-x}+\sqrt {1+x}}\right ) \]
-2*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*(1 - x^2) ^(1/4))/(Sqrt[1 - x] - Sqrt[1 + x])] - 2*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/ 4)] + Sqrt[2]*ArcTanh[(Sqrt[2]*(1 - x^2)^(1/4))/(Sqrt[1 - x] + Sqrt[1 + x] )]
Time = 0.37 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {140, 73, 104, 756, 216, 219, 854, 826, 1476, 1082, 217, 1479, 25, 27, 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.
\(\displaystyle \int \frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x} x} \, dx\) |
\(\Big \downarrow \) 140 |
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x} (x+1)^{3/4}}dx+\int \frac {1}{\sqrt [4]{1-x} x (x+1)^{3/4}}dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x} x (x+1)^{3/4}}dx-4 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle 4 \int \frac {1}{\frac {x+1}{1-x}-1}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-4 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle 4 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {x+1}}{\sqrt {1-x}}}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {x+1}}{\sqrt {1-x}}+1}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-4 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 4 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {x+1}}{\sqrt {1-x}}}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \int \frac {\sqrt {1-x}}{(x+1)^{3/4}}d\sqrt [4]{1-x}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \int \frac {\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \int \frac {\sqrt {1-x}+1}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-x}}{2-x}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}{\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1}d\frac {\sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 4 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-x}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-x}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt {2}}\right )\right )\) |
4*(-1/2*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] - ArcTanh[(1 + x)^(1/4)/(1 - x )^(1/4)]/2) - 4*((-(ArcTan[1 - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt [2]) + ArcTan[1 + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2])/2 + (Log [1 + Sqrt[1 - x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt[2]) - Lo g[1 + Sqrt[1 - x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt[2]))/2)
3.9.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
\[\int \frac {\left (1+x \right )^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}} x}d x\]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x - i - 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x + i - 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x - i + 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x + i + 1\right )} + 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + 2 \, \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) \]
-(1/2*I + 1/2)*sqrt(2)*log((sqrt(2)*((I + 1)*x - I - 1) + 2*(x + 1)^(1/4)* (-x + 1)^(3/4))/(x - 1)) + (1/2*I - 1/2)*sqrt(2)*log((sqrt(2)*(-(I - 1)*x + I - 1) + 2*(x + 1)^(1/4)*(-x + 1)^(3/4))/(x - 1)) - (1/2*I - 1/2)*sqrt(2 )*log((sqrt(2)*((I - 1)*x - I + 1) + 2*(x + 1)^(1/4)*(-x + 1)^(3/4))/(x - 1)) + (1/2*I + 1/2)*sqrt(2)*log((sqrt(2)*(-(I + 1)*x + I + 1) + 2*(x + 1)^ (1/4)*(-x + 1)^(3/4))/(x - 1)) + 2*arctan((x + 1)^(1/4)*(-x + 1)^(3/4)/(x - 1)) + log((x + (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - log(-(x - (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1))
\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=\int \frac {\sqrt [4]{x + 1}}{x \sqrt [4]{1 - x}}\, dx \]
\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx=\int \frac {{\left (x+1\right )}^{1/4}}{x\,{\left (1-x\right )}^{1/4}} \,d x \]