Integrand size = 31, antiderivative size = 233 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right ) \]
arctan(x/(x^4-x^2)^(1/4))-1/2*(2+2*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2) )^(1/2)*x/(x^4-x^2)^(1/4))-1/2*(-2+2*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2 ))^(1/2)*x/(x^4-x^2)^(1/4))+arctanh(x/(x^4-x^2)^(1/4))-1/2*(2+2*5^(1/2))^( 1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-x^2)^(1/4))-1/2*(-2+2*5^(1/2) )^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-x^2)^(1/4))
Time = 0.70 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.08 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt {2 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt {2 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt {2 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-\sqrt {2 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{2 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
(Sqrt[x]*(-1 + x^2)^(1/4)*(2*ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)] - Sqrt[2*(1 + Sqrt[5])]*ArcTan[(Sqrt[(-1 + Sqrt[5])/2]*Sqrt[x])/(-1 + x^2)^(1/4)] - Sq rt[2*(-1 + Sqrt[5])]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*Sqrt[x])/(-1 + x^2)^(1/ 4)] + 2*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)] - Sqrt[2*(1 + Sqrt[5])]*ArcTanh[ (Sqrt[(-1 + Sqrt[5])/2]*Sqrt[x])/(-1 + x^2)^(1/4)] - Sqrt[2*(-1 + Sqrt[5]) ]*ArcTanh[(Sqrt[(1 + Sqrt[5])/2]*Sqrt[x])/(-1 + x^2)^(1/4)]))/(2*(x^2*(-1 + x^2))^(1/4))
Time = 0.86 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2467, 25, 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.
| \(\displaystyle \int \frac {x^4+1}{\left (x^4-x^2-1\right ) \sqrt [4]{x^4-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int -\frac {x^4+1}{\sqrt {x} \sqrt [4]{x^2-1} \left (-x^4+x^2+1\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {x^4+1}{\sqrt {x} \sqrt [4]{x^2-1} \left (-x^4+x^2+1\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {x^4+1}{\sqrt [4]{x^2-1} \left (-x^4+x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {x^2+2}{\sqrt [4]{x^2-1} \left (-x^4+x^2+1\right )}-\frac {1}{\sqrt [4]{x^2-1}}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )\right )}{\sqrt [4]{x^4-x^2}}\) |
(-2*Sqrt[x]*(-1 + x^2)^(1/4)*(-1/2*ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)] + (((3 + Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*Sqrt[x])/(-1 + x^2)^( 1/4)])/2 + (((3 - Sqrt[5])/2)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*Sqrt[x ])/(-1 + x^2)^(1/4)])/2 - ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)]/2 + (((3 + Sqr t[5])/2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)] )/2 + (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*Sqrt[x])/( -1 + x^2)^(1/4)])/2))/(-x^2 + x^4)^(1/4)
3.27.37.3.1 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 = 6.88 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {\left (-\operatorname {arctanh}\left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )+\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {2+2 \sqrt {5}}}\right )\right ) \sqrt {-2+2 \sqrt {5}}}{2}+\frac {\left (\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )-\operatorname {arctanh}\left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x \sqrt {-2+2 \sqrt {5}}}\right )\right ) \sqrt {2+2 \sqrt {5}}}{2}+\frac {\ln \left (\frac {x +\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{2}-\frac {\ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}-x}{x}\right )}{2}-\arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )\) | \(194\) |
1/2*(-arctanh(2*(x^4-x^2)^(1/4)/x/(2+2*5^(1/2))^(1/2))+arctan(2*(x^4-x^2)^ (1/4)/x/(2+2*5^(1/2))^(1/2)))*(-2+2*5^(1/2))^(1/2)+1/2*(arctan(2*(x^4-x^2) ^(1/4)/x/(-2+2*5^(1/2))^(1/2))-arctanh(2*(x^4-x^2)^(1/4)/x/(-2+2*5^(1/2))^ (1/2)))*(2+2*5^(1/2))^(1/2)+1/2*ln((x+(x^4-x^2)^(1/4))/x)-1/2*ln(((x^4-x^2 )^(1/4)-x)/x)-arctan((x^4-x^2)^(1/4)/x)
Leaf count of result is larger than twice the leaf count of optimal. 1689 vs. \(2 (169) = 338\).
Time = 47.33 (sec) , antiderivative size = 1689, normalized size of antiderivative = 7.25 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\text {Too large to display} \]
1/8*sqrt(2)*sqrt(-sqrt(5) + 1)*log((2*sqrt(x^4 - x^2)*(sqrt(5)*sqrt(2)*(18 1*x^3 - 95*x) + sqrt(2)*(9*x^3 - 457*x))*sqrt(-sqrt(5) + 1) + 4*(x^4 - x^2 )^(3/4)*(448*x^2 - sqrt(5)*(86*x^2 + 181) - 9) - (sqrt(5)*sqrt(2)*(9*x^5 - 371*x^3 + 181*x) + sqrt(2)*(905*x^5 - 923*x^3 + 9*x))*sqrt(-sqrt(5) + 1) - 4*(9*x^4 - 457*x^2 + sqrt(5)*(181*x^4 - 95*x^2))*(x^4 - x^2)^(1/4))/(x^5 - x^3 - x)) - 1/8*sqrt(2)*sqrt(-sqrt(5) + 1)*log(-(2*sqrt(x^4 - x^2)*(sqr t(5)*sqrt(2)*(181*x^3 - 95*x) + sqrt(2)*(9*x^3 - 457*x))*sqrt(-sqrt(5) + 1 ) - 4*(x^4 - x^2)^(3/4)*(448*x^2 - sqrt(5)*(86*x^2 + 181) - 9) - (sqrt(5)* sqrt(2)*(9*x^5 - 371*x^3 + 181*x) + sqrt(2)*(905*x^5 - 923*x^3 + 9*x))*sqr t(-sqrt(5) + 1) + 4*(9*x^4 - 457*x^2 + sqrt(5)*(181*x^4 - 95*x^2))*(x^4 - x^2)^(1/4))/(x^5 - x^3 - x)) + 1/8*sqrt(2)*sqrt(-sqrt(5) - 1)*log((2*sqrt( x^4 - x^2)*(sqrt(5)*sqrt(2)*(181*x^3 - 95*x) - sqrt(2)*(9*x^3 - 457*x))*sq rt(-sqrt(5) - 1) + 4*(x^4 - x^2)^(3/4)*(448*x^2 + sqrt(5)*(86*x^2 + 181) - 9) + (sqrt(5)*sqrt(2)*(9*x^5 - 371*x^3 + 181*x) - sqrt(2)*(905*x^5 - 923* x^3 + 9*x))*sqrt(-sqrt(5) - 1) + 4*(9*x^4 - 457*x^2 - sqrt(5)*(181*x^4 - 9 5*x^2))*(x^4 - x^2)^(1/4))/(x^5 - x^3 - x)) - 1/8*sqrt(2)*sqrt(-sqrt(5) - 1)*log(-(2*sqrt(x^4 - x^2)*(sqrt(5)*sqrt(2)*(181*x^3 - 95*x) - sqrt(2)*(9* x^3 - 457*x))*sqrt(-sqrt(5) - 1) - 4*(x^4 - x^2)^(3/4)*(448*x^2 + sqrt(5)* (86*x^2 + 181) - 9) + (sqrt(5)*sqrt(2)*(9*x^5 - 371*x^3 + 181*x) - sqrt(2) *(905*x^5 - 923*x^3 + 9*x))*sqrt(-sqrt(5) - 1) - 4*(9*x^4 - 457*x^2 - s...
\[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {x^{4} + 1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - x^{2} - 1\right )}\, dx \]
\[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} - 1\right )}} \,d x } \]
Time = 0.40 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\frac {1}{2} \, \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 2} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{2} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \]
1/2*sqrt(2*sqrt(5) - 2)*arctan((-1/x^2 + 1)^(1/4)/sqrt(1/2*sqrt(5) + 1/2)) + 1/2*sqrt(2*sqrt(5) + 2)*arctan((-1/x^2 + 1)^(1/4)/sqrt(1/2*sqrt(5) - 1/ 2)) - 1/4*sqrt(2*sqrt(5) - 2)*log(sqrt(1/2*sqrt(5) + 1/2) + (-1/x^2 + 1)^( 1/4)) + 1/4*sqrt(2*sqrt(5) - 2)*log(sqrt(1/2*sqrt(5) + 1/2) - (-1/x^2 + 1) ^(1/4)) - 1/4*sqrt(2*sqrt(5) + 2)*log(sqrt(1/2*sqrt(5) - 1/2) + (-1/x^2 + 1)^(1/4)) + 1/4*sqrt(2*sqrt(5) + 2)*log(abs(-sqrt(1/2*sqrt(5) - 1/2) + (-1 /x^2 + 1)^(1/4))) - arctan((-1/x^2 + 1)^(1/4)) + 1/2*log((-1/x^2 + 1)^(1/4 ) + 1) - 1/2*log(-(-1/x^2 + 1)^(1/4) + 1)
Timed out. \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int -\frac {x^4+1}{{\left (x^4-x^2\right )}^{1/4}\,\left (-x^4+x^2+1\right )} \,d x \]