Integrand size = 34, antiderivative size = 141 \[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=\frac {\sqrt [4]{1+2 x^4} \left (2+9 x^4\right )}{20 x^5}+\frac {9}{8} \sqrt [4]{\frac {3}{2}} \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+2 x^4}}\right )-\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+2 x^4}}\right )-\frac {9}{8} \sqrt [4]{\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+2 x^4}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+2 x^4}}\right ) \]
1/20*(2*x^4+1)^(1/4)*(9*x^4+2)/x^5+9/16*3^(1/4)*2^(3/4)*arctan(1/2*3^(1/4) *2^(3/4)*x/(2*x^4+1)^(1/4))-2^(1/4)*arctan(2^(1/4)*x/(2*x^4+1)^(1/4))-9/16 *3^(1/4)*2^(3/4)*arctanh(1/2*3^(1/4)*2^(3/4)*x/(2*x^4+1)^(1/4))+2^(1/4)*ar ctanh(2^(1/4)*x/(2*x^4+1)^(1/4))
Time = 0.47 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=\frac {\sqrt [4]{1+2 x^4} \left (2+9 x^4\right )}{20 x^5}-\sqrt [4]{2} \arctan \left (\frac {x}{\sqrt [4]{\frac {1}{2}+x^4}}\right )+\frac {9}{8} \sqrt [4]{\frac {3}{2}} \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+2 x^4}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{\frac {1}{2}+x^4}}\right )-\frac {9}{8} \sqrt [4]{\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+2 x^4}}\right ) \]
((1 + 2*x^4)^(1/4)*(2 + 9*x^4))/(20*x^5) - 2^(1/4)*ArcTan[x/(1/2 + x^4)^(1 /4)] + (9*(3/2)^(1/4)*ArcTan[((3/2)^(1/4)*x)/(1 + 2*x^4)^(1/4)])/8 + 2^(1/ 4)*ArcTanh[x/(1/2 + x^4)^(1/4)] - (9*(3/2)^(1/4)*ArcTanh[((3/2)^(1/4)*x)/( 1 + 2*x^4)^(1/4)])/8
Time = 0.41 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1387, 1050, 27, 1050, 25, 1054, 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.
| \(\displaystyle \int \frac {\sqrt [4]{2 x^4+1} \left (2 x^8-x^4-1\right )}{x^6 \left (x^4+2\right )} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {\left (x^4-1\right ) \left (2 x^4+1\right )^{5/4}}{x^6 \left (x^4+2\right )}dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{10} \int -\frac {5 \left (1-4 x^4\right ) \sqrt [4]{2 x^4+1}}{x^2 \left (x^4+2\right )}dx+\frac {\left (2 x^4+1\right )^{5/4}}{10 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (2 x^4+1\right )^{5/4}}{10 x^5}-\frac {1}{2} \int \frac {\left (1-4 x^4\right ) \sqrt [4]{2 x^4+1}}{x^2 \left (x^4+2\right )}dx\) |
| \(\Big \downarrow \) 1050 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt [4]{2 x^4+1}}{2 x}-\frac {1}{2} \int -\frac {x^2 \left (16 x^4+5\right )}{\left (x^4+2\right ) \left (2 x^4+1\right )^{3/4}}dx\right )+\frac {\left (2 x^4+1\right )^{5/4}}{10 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {x^2 \left (16 x^4+5\right )}{\left (x^4+2\right ) \left (2 x^4+1\right )^{3/4}}dx+\frac {\sqrt [4]{2 x^4+1}}{2 x}\right )+\frac {\left (2 x^4+1\right )^{5/4}}{10 x^5}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \left (\frac {16 x^2}{\left (2 x^4+1\right )^{3/4}}-\frac {27 x^2}{\left (x^4+2\right ) \left (2 x^4+1\right )^{3/4}}\right )dx+\frac {\sqrt [4]{2 x^4+1}}{2 x}\right )+\frac {\left (2 x^4+1\right )^{5/4}}{10 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {9}{2} \sqrt [4]{\frac {3}{2}} \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{2 x^4+1}}\right )-4 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )-\frac {9}{2} \sqrt [4]{\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{2 x^4+1}}\right )+4 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4+1}}\right )\right )+\frac {\sqrt [4]{2 x^4+1}}{2 x}\right )+\frac {\left (2 x^4+1\right )^{5/4}}{10 x^5}\) |
(1 + 2*x^4)^(5/4)/(10*x^5) + ((1 + 2*x^4)^(1/4)/(2*x) + ((9*(3/2)^(1/4)*Ar cTan[((3/2)^(1/4)*x)/(1 + 2*x^4)^(1/4)])/2 - 4*2^(1/4)*ArcTan[(2^(1/4)*x)/ (1 + 2*x^4)^(1/4)] - (9*(3/2)^(1/4)*ArcTanh[((3/2)^(1/4)*x)/(1 + 2*x^4)^(1 /4)])/2 + 4*2^(1/4)*ArcTanh[(2^(1/4)*x)/(1 + 2*x^4)^(1/4)])/2)/2
3.21.4.3.1 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[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^n*(m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1 ))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && G tQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(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, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 8.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.34
| method | result | size |
| pseudoelliptic | \(\frac {-45 \ln \left (\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x +2 \left (2 x^{4}+1\right )^{\frac {1}{4}}}{-2^{\frac {3}{4}} 3^{\frac {1}{4}} x +2 \left (2 x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{5}-90 \arctan \left (\frac {3^{\frac {3}{4}} 2^{\frac {1}{4}} \left (2 x^{4}+1\right )^{\frac {1}{4}}}{3 x}\right ) 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{5}+80 \ln \left (\frac {2^{\frac {1}{4}} x +\left (2 x^{4}+1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (2 x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} x^{5}+160 \arctan \left (\frac {2^{\frac {3}{4}} \left (2 x^{4}+1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}} x^{5}+72 \left (2 x^{4}+1\right )^{\frac {1}{4}} x^{4}+16 \left (2 x^{4}+1\right )^{\frac {1}{4}}}{160 x^{5}}\) | \(189\) |
1/160*(-45*ln((2^(3/4)*3^(1/4)*x+2*(2*x^4+1)^(1/4))/(-2^(3/4)*3^(1/4)*x+2* (2*x^4+1)^(1/4)))*2^(3/4)*3^(1/4)*x^5-90*arctan(1/3*3^(3/4)*2^(1/4)/x*(2*x ^4+1)^(1/4))*2^(3/4)*3^(1/4)*x^5+80*ln((2^(1/4)*x+(2*x^4+1)^(1/4))/(-2^(1/ 4)*x+(2*x^4+1)^(1/4)))*2^(1/4)*x^5+160*arctan(1/2*2^(3/4)/x*(2*x^4+1)^(1/4 ))*2^(1/4)*x^5+72*(2*x^4+1)^(1/4)*x^4+16*(2*x^4+1)^(1/4))/x^5
Result contains complex when optimal does not.
Time = 20.39 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.58 \[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=-\frac {45 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (7 \, x^{4} + 2\right )} + 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 45 i \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {6 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 i \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} - 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (7 i \, x^{4} + 2 i\right )} - 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) + 45 i \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {6 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 i \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} - 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (-7 i \, x^{4} - 2 i\right )} - 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 45 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (\frac {6 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {2 \, x^{4} + 1} x^{2} - 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (7 \, x^{4} + 2\right )} + 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 80 \cdot 2^{\frac {1}{4}} x^{5} \log \left (4 \, \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 4 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x\right ) + 80 \cdot 2^{\frac {1}{4}} x^{5} \log \left (4 \, \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 4 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x\right ) - 80 i \cdot 2^{\frac {1}{4}} x^{5} \log \left (-4 \, \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (-4 i \, x^{4} - i\right )} + 4 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x\right ) + 80 i \cdot 2^{\frac {1}{4}} x^{5} \log \left (-4 \, \sqrt {2} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (4 i \, x^{4} + i\right )} + 4 \, {\left (2 \, x^{4} + 1\right )}^{\frac {3}{4}} x\right ) - 16 \, {\left (9 \, x^{4} + 2\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{320 \, x^{5}} \]
-1/320*(45*3^(1/4)*2^(3/4)*x^5*log((6*sqrt(3)*sqrt(2)*(2*x^4 + 1)^(1/4)*x^ 3 + 6*3^(1/4)*2^(3/4)*sqrt(2*x^4 + 1)*x^2 + 3^(3/4)*2^(1/4)*(7*x^4 + 2) + 12*(2*x^4 + 1)^(3/4)*x)/(x^4 + 2)) - 45*I*3^(1/4)*2^(3/4)*x^5*log(-(6*sqrt (3)*sqrt(2)*(2*x^4 + 1)^(1/4)*x^3 + 6*I*3^(1/4)*2^(3/4)*sqrt(2*x^4 + 1)*x^ 2 - 3^(3/4)*2^(1/4)*(7*I*x^4 + 2*I) - 12*(2*x^4 + 1)^(3/4)*x)/(x^4 + 2)) + 45*I*3^(1/4)*2^(3/4)*x^5*log(-(6*sqrt(3)*sqrt(2)*(2*x^4 + 1)^(1/4)*x^3 - 6*I*3^(1/4)*2^(3/4)*sqrt(2*x^4 + 1)*x^2 - 3^(3/4)*2^(1/4)*(-7*I*x^4 - 2*I) - 12*(2*x^4 + 1)^(3/4)*x)/(x^4 + 2)) - 45*3^(1/4)*2^(3/4)*x^5*log((6*sqrt (3)*sqrt(2)*(2*x^4 + 1)^(1/4)*x^3 - 6*3^(1/4)*2^(3/4)*sqrt(2*x^4 + 1)*x^2 - 3^(3/4)*2^(1/4)*(7*x^4 + 2) + 12*(2*x^4 + 1)^(3/4)*x)/(x^4 + 2)) - 80*2^ (1/4)*x^5*log(4*sqrt(2)*(2*x^4 + 1)^(1/4)*x^3 + 4*2^(1/4)*sqrt(2*x^4 + 1)* x^2 + 2^(3/4)*(4*x^4 + 1) + 4*(2*x^4 + 1)^(3/4)*x) + 80*2^(1/4)*x^5*log(4* sqrt(2)*(2*x^4 + 1)^(1/4)*x^3 - 4*2^(1/4)*sqrt(2*x^4 + 1)*x^2 - 2^(3/4)*(4 *x^4 + 1) + 4*(2*x^4 + 1)^(3/4)*x) - 80*I*2^(1/4)*x^5*log(-4*sqrt(2)*(2*x^ 4 + 1)^(1/4)*x^3 + 4*I*2^(1/4)*sqrt(2*x^4 + 1)*x^2 + 2^(3/4)*(-4*I*x^4 - I ) + 4*(2*x^4 + 1)^(3/4)*x) + 80*I*2^(1/4)*x^5*log(-4*sqrt(2)*(2*x^4 + 1)^( 1/4)*x^3 - 4*I*2^(1/4)*sqrt(2*x^4 + 1)*x^2 + 2^(3/4)*(4*I*x^4 + I) + 4*(2* x^4 + 1)^(3/4)*x) - 16*(9*x^4 + 2)*(2*x^4 + 1)^(1/4))/x^5
\[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (2 x^{4} + 1\right )^{\frac {5}{4}}}{x^{6} \left (x^{4} + 2\right )}\, dx \]
\[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{8} - x^{4} - 1\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{{\left (x^{4} + 2\right )} x^{6}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=-\frac {1}{16} \cdot 54^{\frac {3}{4}} \arctan \left (\frac {24^{\frac {3}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{12 \, x}\right ) - \frac {1}{32} \cdot 54^{\frac {3}{4}} \log \left (\frac {1}{2} \cdot 24^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{32} \cdot 54^{\frac {3}{4}} \log \left (-\frac {1}{2} \cdot 24^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} + 2\right )}}{10 \, x} + \frac {{\left (2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x} \]
-1/16*54^(3/4)*arctan(1/12*24^(3/4)*(2*x^4 + 1)^(1/4)/x) - 1/32*54^(3/4)*l og(1/2*24^(1/4) + (2*x^4 + 1)^(1/4)/x) + 1/32*54^(3/4)*log(-1/2*24^(1/4) + (2*x^4 + 1)^(1/4)/x) + 2^(1/4)*arctan(1/2*2^(3/4)*(2*x^4 + 1)^(1/4)/x) + 1/2*2^(1/4)*log(2^(1/4) + (2*x^4 + 1)^(1/4)/x) - 1/2*2^(1/4)*log(-2^(1/4) + (2*x^4 + 1)^(1/4)/x) + 1/10*(2*x^4 + 1)^(1/4)*(1/x^4 + 2)/x + 1/4*(2*x^4 + 1)^(1/4)/x
Timed out. \[ \int \frac {\sqrt [4]{1+2 x^4} \left (-1-x^4+2 x^8\right )}{x^6 \left (2+x^4\right )} \, dx=\int -\frac {{\left (2\,x^4+1\right )}^{1/4}\,\left (-2\,x^8+x^4+1\right )}{x^6\,\left (x^4+2\right )} \,d x \]