Integrand size = 22, antiderivative size = 162 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{4\ 2^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{4\ 2^{3/4}} \]
-1/8*arctan(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)-1/8*arctan(2^(3/4)*x*(x^6+x ^2)^(1/4)/(2^(1/2)*x^2-(x^6+x^2)^(1/2)))*2^(1/4)-1/8*arctanh(2^(1/4)*x/(x^ 6+x^2)^(1/4))*2^(3/4)+1/8*arctanh((1/2*x^2*2^(3/4)+1/2*(x^6+x^2)^(1/2)*2^( 1/4))/x/(x^6+x^2)^(1/4))*2^(1/4)
Time = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {\sqrt {x} \sqrt [4]{1+x^4} \left (\sqrt {2} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{4\ 2^{3/4} \sqrt [4]{x^2+x^6}} \]
-1/4*(Sqrt[x]*(1 + x^4)^(1/4)*(Sqrt[2]*ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^4)^ (1/4)] + ArcTan[(2^(3/4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^ 4])] + Sqrt[2]*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] - ArcTanh[(2*2^( 1/4)*Sqrt[x]*(1 + x^4)^(1/4))/(2*x + Sqrt[2]*Sqrt[1 + x^4])]))/(2^(3/4)*(x ^2 + x^6)^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.28, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1948, 25, 966, 1012}
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.
| \(\displaystyle \int \frac {x^2}{\left (x^4-1\right ) \sqrt [4]{x^6+x^2}} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^4+1} \int -\frac {x^{3/2}}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^{3/2}}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}dx}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 966 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^4+1} \int \frac {x^2}{\left (1-x^4\right ) \sqrt [4]{x^4+1}}d\sqrt {x}}{\sqrt [4]{x^6+x^2}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle -\frac {2 x^3 \sqrt [4]{x^4+1} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},x^4,-x^4\right )}{5 \sqrt [4]{x^6+x^2}}\) |
3.22.84.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 19.06 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.26
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}-2 \arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) \sqrt {2}+\ln \left (\frac {-2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{2^{\frac {3}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}+x}{x}\right )\right )}{16}\) | \(204\) |
| trager | \(\text {Expression too large to display}\) | \(638\) |
-1/16*2^(1/4)*(ln((-2^(1/4)*x-(x^2*(x^4+1))^(1/4))/(2^(1/4)*x-(x^2*(x^4+1) )^(1/4)))*2^(1/2)-2*arctan(1/2*(x^2*(x^4+1))^(1/4)/x*2^(3/4))*2^(1/2)+ln(( -2^(3/4)*(x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1/2))/(2^(3/4)*( x^2*(x^4+1))^(1/4)*x+2^(1/2)*x^2+(x^2*(x^4+1))^(1/2)))+2*arctan((2^(1/4)*( x^2*(x^4+1))^(1/4)-x)/x)+2*arctan((2^(1/4)*(x^2*(x^4+1))^(1/4)+x)/x))
Result contains complex when optimal does not.
Time = 8.28 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.93 \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + \frac {1}{32} \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) + \frac {1}{32} i \cdot 2^{\frac {3}{4}} \log \left (\frac {4 i \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2 i \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - \frac {1}{32} i \cdot 2^{\frac {3}{4}} \log \left (\frac {-4 i \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 i \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x}{x^{5} - 2 \, x^{3} + x}\right ) - \left (\frac {1}{32} i - \frac {1}{32}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (-i \, x^{5} + 2 i \, x^{3} - i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x - \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) + \left (\frac {1}{32} i + \frac {1}{32}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {-\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (i \, x^{5} - 2 i \, x^{3} + i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) - \left (\frac {1}{32} i + \frac {1}{32}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {\left (2 i - 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (i \, x^{5} - 2 i \, x^{3} + i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x - \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) + \left (\frac {1}{32} i - \frac {1}{32}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {-\left (2 i + 2\right ) \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (-i \, x^{5} + 2 i \, x^{3} - i \, x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}\right ) \]
-1/32*2^(3/4)*log((4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2) ^(3/4) + sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x )) + 1/32*2^(3/4)*log(-(4*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) - 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^ 3 + x)) + 1/32*I*2^(3/4)*log((4*I*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 - 2*I*2^(3 /4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/( x^5 - 2*x^3 + x)) - 1/32*I*2^(3/4)*log((-4*I*2^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2*I*2^(3/4)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) + 4*sqrt(x^6 + x^2)*x)/(x^5 - 2*x^3 + x)) - (1/32*I - 1/32)*2^(1/4)*log(((2*I + 2)*2^(3 /4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(-I*x^5 + 2*I*x^3 - I*x) + 4*sqrt(x^6 + x^2)*x - (2*I - 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + (1/32 *I + 1/32)*2^(1/4)*log((-(2*I - 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2) *(I*x^5 - 2*I*x^3 + I*x) + 4*sqrt(x^6 + x^2)*x + (2*I + 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - (1/32*I + 1/32)*2^(1/4)*log(((2*I - 2)*2^ (3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(I*x^5 - 2*I*x^3 + I*x) + 4*sqrt(x^6 + x^2)*x - (2*I + 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + (1/3 2*I - 1/32)*2^(1/4)*log((-(2*I + 2)*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2 )*(-I*x^5 + 2*I*x^3 - I*x) + 4*sqrt(x^6 + x^2)*x + (2*I - 2)*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x))
\[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {x^{2}}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{2}}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
\[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{2}}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {x^2}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )} \,d x \]