Integrand size = 28, antiderivative size = 122 \[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=-\frac {3 \left (-x^2+x^4\right )^{3/4}}{x \left (-1+x^2\right )}+2 \arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}}+2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^2+x^4}}\right )}{2 \sqrt [4]{2}} \]
-3*(x^4-x^2)^(3/4)/x/(x^2-1)+2*arctan(x/(x^4-x^2)^(1/4))-3/4*arctan(2^(1/4 )*x/(x^4-x^2)^(1/4))*2^(3/4)+2*arctanh(x/(x^4-x^2)^(1/4))-3/4*arctanh(2^(1 /4)*x/(x^4-x^2)^(1/4))*2^(3/4)
Time = 0.38 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.30 \[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=-\frac {\sqrt {x} \left (12 \sqrt {x}-8 \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+3\ 2^{3/4} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-8 \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+3\ 2^{3/4} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \]
-1/4*(Sqrt[x]*(12*Sqrt[x] - 8*(-1 + x^2)^(1/4)*ArcTan[Sqrt[x]/(-1 + x^2)^( 1/4)] + 3*2^(3/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/ 4)] - 8*(-1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)] + 3*2^(3/4)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)]))/(x^2*(-1 + x^2 ))^(1/4)
Time = 0.65 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2467, 25, 1388, 2035, 25, 7276, 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.
| \(\displaystyle \int \frac {2 x^4+1}{\left (x^4-1\right ) \sqrt [4]{x^4-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int -\frac {2 x^4+1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^4\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {2 x^4+1}{\sqrt {x} \sqrt [4]{x^2-1} \left (1-x^4\right )}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {2 x^4+1}{\sqrt {x} \left (-x^2-1\right ) \left (x^2-1\right )^{5/4}}dx}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int -\frac {2 x^4+1}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {2 x^4+1}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {2 x^2}{\left (x^2-1\right )^{5/4}}+\frac {3}{\left (x^2-1\right )^{5/4} \left (x^2+1\right )}-\frac {2}{\left (x^2-1\right )^{5/4}}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \left (-\arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {3 \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2}}-\text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{2}}+\frac {3 \sqrt {x}}{2 \sqrt [4]{x^2-1}}\right )}{\sqrt [4]{x^4-x^2}}\) |
(-2*Sqrt[x]*(-1 + x^2)^(1/4)*((3*Sqrt[x])/(2*(-1 + x^2)^(1/4)) - ArcTan[Sq rt[x]/(-1 + x^2)^(1/4)] + (3*ArcTan[(2^(1/4)*Sqrt[x])/(-1 + x^2)^(1/4)])/( 4*2^(1/4)) - ArcTanh[Sqrt[x]/(-1 + x^2)^(1/4)] + (3*ArcTanh[(2^(1/4)*Sqrt[ x])/(-1 + x^2)^(1/4)])/(4*2^(1/4))))/(-x^2 + x^4)^(1/4)
3.18.100.3.1 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_)), 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 = 2.40 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.32
| method | result | size |
| pseudoelliptic | \(\frac {\frac {3 \left (\left (\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2 x}\right )-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\right )}{2}\right ) 2^{\frac {3}{4}}-\frac {8 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{3}+\frac {4 \ln \left (\frac {x +\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{3}-\frac {4 \ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{4}}-x}{x}\right )}{3}\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{4}-3 x}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(161\) |
3/4*(((arctan(1/2*2^(3/4)/x*(x^4-x^2)^(1/4))-1/2*ln((-2^(1/4)*x-(x^4-x^2)^ (1/4))/(2^(1/4)*x-(x^4-x^2)^(1/4))))*2^(3/4)-8/3*arctan(1/x*(x^4-x^2)^(1/4 ))+4/3*ln((x+(x^4-x^2)^(1/4))/x)-4/3*ln(((x^4-x^2)^(1/4)-x)/x))*(x^4-x^2)^ (1/4)-4*x)/(x^4-x^2)^(1/4)
Result contains complex when optimal does not.
Time = 10.77 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.92 \[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=-\frac {3 \cdot 2^{\frac {3}{4}} {\left (x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) - 3 \cdot 2^{\frac {3}{4}} {\left (x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} - x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x + 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 3 \cdot 2^{\frac {3}{4}} {\left (i \, x^{3} - i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{3} - i \, x\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 3 \cdot 2^{\frac {3}{4}} {\left (-i \, x^{3} + i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{3} + i \, x\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - x^{2}} x - 4 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x}\right ) + 16 \, {\left (x^{3} - x\right )} \arctan \left (\frac {2 \, {\left ({\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) - 16 \, {\left (x^{3} - x\right )} \log \left (\frac {2 \, x^{3} + 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x^{2}} x - x + 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x}\right ) + 48 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{16 \, {\left (x^{3} - x\right )}} \]
-1/16*(3*2^(3/4)*(x^3 - x)*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 + 2^(3/4)* (3*x^3 - x) + 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x) ) - 3*2^(3/4)*(x^3 - x)*log((4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(3* x^3 - x) - 4*2^(1/4)*sqrt(x^4 - x^2)*x + 4*(x^4 - x^2)^(3/4))/(x^3 + x)) + 3*2^(3/4)*(I*x^3 - I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*( 3*I*x^3 - I*x) + 4*I*2^(1/4)*sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^(3/4))/(x^3 + x)) + 3*2^(3/4)*(-I*x^3 + I*x)*log(-(4*sqrt(2)*(x^4 - x^2)^(1/4)*x^2 - 2^(3/4)*(-3*I*x^3 + I*x) - 4*I*2^(1/4)*sqrt(x^4 - x^2)*x - 4*(x^4 - x^2)^( 3/4))/(x^3 + x)) + 16*(x^3 - x)*arctan(2*((x^4 - x^2)^(1/4)*x^2 + (x^4 - x ^2)^(3/4))/x) - 16*(x^3 - x)*log((2*x^3 + 2*(x^4 - x^2)^(1/4)*x^2 + 2*sqrt (x^4 - x^2)*x - x + 2*(x^4 - x^2)^(3/4))/x) + 48*(x^4 - x^2)^(3/4))/(x^3 - x)
\[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {2 x^{4} + 1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int { \frac {2 \, x^{4} + 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\frac {3}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {3}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} - 2 \, \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \]
3/4*2^(3/4)*arctan(1/2*2^(3/4)*(-1/x^2 + 1)^(1/4)) - 3/8*2^(3/4)*log(2^(1/ 4) + (-1/x^2 + 1)^(1/4)) + 3/8*2^(3/4)*log(2^(1/4) - (-1/x^2 + 1)^(1/4)) - 3/(-1/x^2 + 1)^(1/4) - 2*arctan((-1/x^2 + 1)^(1/4)) + log((-1/x^2 + 1)^(1 /4) + 1) - log(-(-1/x^2 + 1)^(1/4) + 1)
Timed out. \[ \int \frac {1+2 x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {2\,x^4+1}{\left (x^4-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \]