Integrand size = 29, antiderivative size = 62 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.50 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {\left (x^2+x^4\right )^{3/4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-3+2 \text {$\#$1}^4}\&\right ]}{2 x^{3/2} \left (1+x^2\right )^{3/4}} \]
-1/2*((x^2 + x^4)^(3/4)*RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[Sqrt[x]]*#1^3 ) + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1]*#1^3)/(-3 + 2*#1^4) & ])/(x^(3/2)*(1 + x^2)^(3/4))
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 522, normalized size of antiderivative = 8.42, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2467, 1592, 1758, 25, 916, 770, 756, 216, 219, 902, 756, 218, 221}
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.
| \(\displaystyle \int \frac {x^2+1}{\left (x^4-x^2+1\right ) \sqrt [4]{x^4+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4}}{\sqrt {x} \left (x^4-x^2+1\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1592 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {\left (x^2+1\right )^{3/4}}{x^4-x^2+1}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1758 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \int -\frac {\left (x^2+1\right )^{3/4}}{-2 x^2-i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}-\frac {2 i \int -\frac {\left (x^2+1\right )^{3/4}}{-2 x^2+i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \int \frac {\left (x^2+1\right )^{3/4}}{-2 x^2+i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}-\frac {2 i \int \frac {\left (x^2+1\right )^{3/4}}{-2 x^2-i \sqrt {3}+1}d\sqrt {x}}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 916 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (-\frac {1}{2} \int \frac {1}{\sqrt [4]{x^2+1}}d\sqrt {x}+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{2} \int \frac {1}{\sqrt [4]{x^2+1}}d\sqrt {x}+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (-\frac {1}{2} \int \frac {1}{1-x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (-\frac {1}{2} \int \frac {1}{1-x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \int \frac {1}{x+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \int \frac {1}{x+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2+i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (-2 x^2-i \sqrt {3}+1\right ) \sqrt [4]{x^2+1}}d\sqrt {x}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \int \frac {1}{-\left (\left (3+i \sqrt {3}\right ) x^2\right )+i \sqrt {3}+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \int \frac {1}{-\left (\left (3-i \sqrt {3}\right ) x^2\right )-i \sqrt {3}+1}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {-\sqrt {3}+i}}+\frac {i \int \frac {1}{\sqrt {3 i-\sqrt {3}} x+\sqrt {i-\sqrt {3}}}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {-\sqrt {3}+i}}\right )\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {\sqrt {3}+i}}+\frac {i \int \frac {1}{\sqrt {3 i+\sqrt {3}} x+\sqrt {i+\sqrt {3}}}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {\sqrt {3}+i}}\right )\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i-\sqrt {3}}-\sqrt {3 i-\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {-\sqrt {3}+i}}+\frac {i \sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (-\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \left (\frac {i \int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {\sqrt {3}+i}}+\frac {i \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3+i \sqrt {3}\right ) \left (\frac {i \sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (-\sqrt {3}+i\right )}+\frac {i \sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (-\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}-\frac {2 i \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )\right )+\frac {1}{2} \left (3-i \sqrt {3}\right ) \left (\frac {i \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (\sqrt {3}+i\right )}+\frac {i \sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [4]{x^2+1}}\right )}{2 \left (\sqrt {3}+i\right )}\right )\right )}{\sqrt {3}}\right )}{\sqrt [4]{x^4+x^2}}\) |
(2*Sqrt[x]*(1 + x^2)^(1/4)*(((2*I)*((-1/2*ArcTan[Sqrt[x]/(1 + x^2)^(1/4)] - ArcTanh[Sqrt[x]/(1 + x^2)^(1/4)]/2)/2 + ((3 + I*Sqrt[3])*(((I/2)*((I - S qrt[3])/(3*I - Sqrt[3]))^(1/4)*ArcTan[Sqrt[x]/(((I - Sqrt[3])/(3*I - Sqrt[ 3]))^(1/4)*(1 + x^2)^(1/4))])/(I - Sqrt[3]) + ((I/2)*((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/4)*ArcTanh[Sqrt[x]/(((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/4)*(1 + x^2)^(1/4))])/(I - Sqrt[3])))/2))/Sqrt[3] - ((2*I)*((-1/2*ArcTan[Sqrt[x ]/(1 + x^2)^(1/4)] - ArcTanh[Sqrt[x]/(1 + x^2)^(1/4)]/2)/2 + ((3 - I*Sqrt[ 3])*(((I/2)*((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*ArcTan[Sqrt[x]/(((I + Sq rt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x^2)^(1/4))])/(I + Sqrt[3]) + ((I/2)*(( I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*ArcTanh[Sqrt[x]/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/4)*(1 + x^2)^(1/4))])/(I + Sqrt[3])))/2))/Sqrt[3]))/(x^2 + x ^4)^(1/4)
3.9.15.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[b/d Int[(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b* x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[ Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*(x^( 4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x ] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r) Int[(d + e*x ^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r) Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
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]
Time = 35.84 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{4}-3}\right )}{2}\) | \(51\) |
| trager | \(\text {Expression too large to display}\) | \(1627\) |
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 2.86 (sec) , antiderivative size = 1566, normalized size of antiderivative = 25.26 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \]
(1/24*I + 1/24)*3^(7/8)*sqrt(2)*(-1)^(1/8)*log((3^(7/8)*sqrt(2)*(-1)^(1/8) *((35*I + 35)*x^5 - (17*I + 17)*x^3 - (13*I + 13)*x) - 3*3^(3/8)*sqrt(2)*( -1)^(5/8)*(-(7*I + 7)*x^5 - (19*I + 19)*x^3 - (3*I + 3)*x) - 12*(x^4 + x^2 )^(3/4)*(2*x^2 + sqrt(3)*(-8*I*x^2 + 5*I) + 11) - 6*sqrt(x^4 + x^2)*(3^(5/ 8)*sqrt(2)*(-1)^(3/8)*((3*I - 3)*x^3 - (8*I - 8)*x) + 3^(1/8)*sqrt(2)*(-1) ^(7/8)*((13*I - 13)*x^3 - (2*I - 2)*x)) - 12*(x^4 + x^2)^(1/4)*(3^(1/4)*(- 1)^(3/4)*(11*I*x^4 - 13*I*x^2) + 3^(3/4)*(-1)^(1/4)*(-5*I*x^4 - 3*I*x^2))) /(x^5 - x^3 + x)) - (1/24*I - 1/24)*3^(7/8)*sqrt(2)*(-1)^(1/8)*log((3^(7/8 )*sqrt(2)*(-1)^(1/8)*(-(35*I - 35)*x^5 + (17*I - 17)*x^3 + (13*I - 13)*x) - 3*3^(3/8)*sqrt(2)*(-1)^(5/8)*((7*I - 7)*x^5 + (19*I - 19)*x^3 + (3*I - 3 )*x) - 12*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(-8*I*x^2 + 5*I) + 11) - 6*sq rt(x^4 + x^2)*(3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(3*I + 3)*x^3 + (8*I + 8)*x) + 3^(1/8)*sqrt(2)*(-1)^(7/8)*(-(13*I + 13)*x^3 + (2*I + 2)*x)) - 12*(x^4 + x^2)^(1/4)*(3^(3/4)*(-1)^(1/4)*(5*I*x^4 + 3*I*x^2) + 3^(1/4)*(-1)^(3/4)*(- 11*I*x^4 + 13*I*x^2)))/(x^5 - x^3 + x)) - (1/24*I + 1/24)*3^(7/8)*sqrt(2)* (-1)^(1/8)*log(-(3*3^(3/8)*sqrt(2)*(-1)^(5/8)*((7*I + 7)*x^5 + (19*I + 19) *x^3 + (3*I + 3)*x) - 3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(35*I + 35)*x^5 + (17*I + 17)*x^3 + (13*I + 13)*x) + 12*(x^4 + x^2)^(3/4)*(2*x^2 + sqrt(3)*(-8*I* x^2 + 5*I) + 11) + 6*sqrt(x^4 + x^2)*(3^(1/8)*sqrt(2)*(-1)^(7/8)*(-(13*I - 13)*x^3 + (2*I - 2)*x) + 3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(3*I - 3)*x^3 + ...
Not integrable
Time = 2.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.42 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{2} + 1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.74 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}} \,d x } \]
2/21*(8*x^5 - 7*(x^3 + x)*x^2 + 9*x^3 + x)/((x^(9/2) - x^(5/2) + sqrt(x))* (x^2 + 1)^(1/4)) + integrate(-4/21*(16*x^4 - 8*(x^4 + 2*x^2 + 1)*x^2 + 11* x^2 - 5)/((x^(17/2) - 2*x^(13/2) + 3*x^(9/2) - 2*x^(5/2) + sqrt(x))*(x^2 + 1)^(1/4)), x)
Not integrable
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - x^{2} + 1\right )}} \,d x } \]
Not integrable
Time = 5.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1+x^2}{\left (1-x^2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^2+1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-x^2+1\right )} \,d x \]