Integrand size = 29, antiderivative size = 117 \[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=\frac {1}{192} \sqrt [4]{x^2+x^4} \left (-7 x+4 x^3+32 x^5\right )-\frac {7}{128} \arctan \left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}}+\frac {7}{128} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2^{3/4}} \]
1/192*(x^4+x^2)^(1/4)*(32*x^5+4*x^3-7*x)-7/128*arctan(x/(x^4+x^2)^(1/4))-1 /2*arctan(2^(1/4)*x/(x^4+x^2)^(1/4))*2^(1/4)+7/128*arctanh(x/(x^4+x^2)^(1/ 4))+1/2*arctanh(2^(1/4)*x/(x^4+x^2)^(1/4))*2^(1/4)
Time = 0.50 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=\frac {\sqrt [4]{x^2+x^4} \left (-14 x^{3/2} \sqrt [4]{1+x^2}+8 x^{7/2} \sqrt [4]{1+x^2}+64 x^{11/2} \sqrt [4]{1+x^2}-21 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )-192 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )+21 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )+192 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{384 \sqrt {x} \sqrt [4]{1+x^2}} \]
((x^2 + x^4)^(1/4)*(-14*x^(3/2)*(1 + x^2)^(1/4) + 8*x^(7/2)*(1 + x^2)^(1/4 ) + 64*x^(11/2)*(1 + x^2)^(1/4) - 21*ArcTan[Sqrt[x]/(1 + x^2)^(1/4)] - 192 *2^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)] + 21*ArcTanh[Sqrt[x]/(1 + x^2)^(1/4)] + 192*2^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)]))/ (384*Sqrt[x]*(1 + x^2)^(1/4))
Time = 0.84 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2467, 1388, 2035, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4+x^2} \left (x^8-x^4-1\right )}{x^4-1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^2} \int \frac {\sqrt {x} \sqrt [4]{x^2+1} \left (-x^8+x^4+1\right )}{1-x^4}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^2} \int \frac {\sqrt {x} \left (-x^8+x^4+1\right )}{\left (1-x^2\right ) \left (x^2+1\right )^{3/4}}dx}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4+x^2} \int \frac {x \left (-x^8+x^4+1\right )}{\left (1-x^2\right ) \left (x^2+1\right )^{3/4}}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4+x^2} \int \left (\frac {x^7}{\left (x^2+1\right )^{3/4}}+\frac {x^5}{\left (x^2+1\right )^{3/4}}+\frac {x}{\left (1-x^2\right ) \left (x^2+1\right )^{3/4}}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4+x^2} \left (-\frac {7}{256} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4}}+\frac {7}{256} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4}}+\frac {1}{12} \sqrt [4]{x^2+1} x^{11/2}+\frac {1}{96} \sqrt [4]{x^2+1} x^{7/2}-\frac {7}{384} \sqrt [4]{x^2+1} x^{3/2}\right )}{\sqrt {x} \sqrt [4]{x^2+1}}\) |
(2*(x^2 + x^4)^(1/4)*((-7*x^(3/2)*(1 + x^2)^(1/4))/384 + (x^(7/2)*(1 + x^2 )^(1/4))/96 + (x^(11/2)*(1 + x^2)^(1/4))/12 - (7*ArcTan[Sqrt[x]/(1 + x^2)^ (1/4)])/256 - ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1/4)]/(2*2^(3/4)) + (7*A rcTanh[Sqrt[x]/(1 + x^2)^(1/4)])/256 + ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2) ^(1/4)]/(2*2^(3/4))))/(Sqrt[x]*(1 + x^2)^(1/4))
3.18.40.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs. \(2(94)=188\).
Time = 3.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.01
| method | result | size |
| pseudoelliptic | \(\frac {x^{6} \left (128 x^{5} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+16 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} x^{3}+192 \,2^{\frac {1}{4}} \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right )+384 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right )-28 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} x +21 \ln \left (\frac {x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )-21 \ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )+42 \arctan \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )\right )}{768 {\left (x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}\right )}^{3} {\left (\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x \right )}^{3} \left (x^{2}+\sqrt {x^{2} \left (x^{2}+1\right )}\right )^{3}}\) | \(235\) |
1/768*x^6*(128*x^5*(x^2*(x^2+1))^(1/4)+16*(x^2*(x^2+1))^(1/4)*x^3+192*2^(1 /4)*ln((-2^(1/4)*x-(x^2*(x^2+1))^(1/4))/(2^(1/4)*x-(x^2*(x^2+1))^(1/4)))+3 84*2^(1/4)*arctan(1/2*(x^2*(x^2+1))^(1/4)/x*2^(3/4))-28*(x^2*(x^2+1))^(1/4 )*x+21*ln((x+(x^2*(x^2+1))^(1/4))/x)-21*ln(((x^2*(x^2+1))^(1/4)-x)/x)+42*a rctan((x^2*(x^2+1))^(1/4)/x))/(x+(x^2*(x^2+1))^(1/4))^3/((x^2*(x^2+1))^(1/ 4)-x)^3/(x^2+(x^2*(x^2+1))^(1/2))^3
Result contains complex when optimal does not.
Time = 4.81 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.42 \[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=\frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x + 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - \frac {1}{32} i \cdot 8^{\frac {3}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + i \cdot 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (3 i \, x^{3} + i \, x\right )} - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + \frac {1}{32} i \cdot 8^{\frac {3}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - i \cdot 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (-3 i \, x^{3} - i \, x\right )} - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x - 8^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + \frac {1}{192} \, {\left (32 \, x^{5} + 4 \, x^{3} - 7 \, x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} - \frac {7}{256} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {7}{256} \, \log \left (\frac {2 \, x^{3} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} + x^{2}} x + x + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \]
1/32*8^(3/4)*log((4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 + 8^(3/4)*sqrt(x^4 + x^2 )*x + 8^(1/4)*(3*x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 1/32*I*8^(3/ 4)*log(-(4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 + I*8^(3/4)*sqrt(x^4 + x^2)*x - 8 ^(1/4)*(3*I*x^3 + I*x) - 4*(x^4 + x^2)^(3/4))/(x^3 - x)) + 1/32*I*8^(3/4)* log(-(4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 - I*8^(3/4)*sqrt(x^4 + x^2)*x - 8^(1 /4)*(-3*I*x^3 - I*x) - 4*(x^4 + x^2)^(3/4))/(x^3 - x)) - 1/32*8^(3/4)*log( (4*sqrt(2)*(x^4 + x^2)^(1/4)*x^2 - 8^(3/4)*sqrt(x^4 + x^2)*x - 8^(1/4)*(3* x^3 + x) + 4*(x^4 + x^2)^(3/4))/(x^3 - x)) + 1/192*(32*x^5 + 4*x^3 - 7*x)* (x^4 + x^2)^(1/4) - 7/256*arctan(2*((x^4 + x^2)^(1/4)*x^2 + (x^4 + x^2)^(3 /4))/x) + 7/256*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)
\[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{8} - x^{4} - 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=\int { \frac {{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}}{x^{4} - 1} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=-\frac {1}{192} \, {\left (7 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {9}{4}} - 18 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {7}{128} \, \arctan \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{256} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{256} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1\right ) \]
-1/192*(7*(1/x^2 + 1)^(9/4) - 18*(1/x^2 + 1)^(5/4) - 21*(1/x^2 + 1)^(1/4)) *x^6 + 1/2*2^(1/4)*arctan(1/2*2^(3/4)*(1/x^2 + 1)^(1/4)) + 1/4*2^(1/4)*log (2^(1/4) + (1/x^2 + 1)^(1/4)) - 1/4*2^(1/4)*log(abs(-2^(1/4) + (1/x^2 + 1) ^(1/4))) + 7/128*arctan((1/x^2 + 1)^(1/4)) + 7/256*log((1/x^2 + 1)^(1/4) + 1) - 7/256*log((1/x^2 + 1)^(1/4) - 1)
Timed out. \[ \int \frac {\sqrt [4]{x^2+x^4} \left (-1-x^4+x^8\right )}{-1+x^4} \, dx=\int -\frac {{\left (x^4+x^2\right )}^{1/4}\,\left (-x^8+x^4+1\right )}{x^4-1} \,d x \]