Integrand size = 16, antiderivative size = 227 \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a+\sqrt {1+a^2}}-\sqrt {2} x}{\sqrt {a+\sqrt {1+a^2}}}\right )}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {a+\sqrt {1+a^2}}}+\frac {\arctan \left (\frac {\sqrt {-a+\sqrt {1+a^2}}+\sqrt {2} x}{\sqrt {a+\sqrt {1+a^2}}}\right )}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {a+\sqrt {1+a^2}}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} x}{\sqrt {1+a^2}+x^2}\right )}{2 \sqrt {2} \sqrt {1+a^2} \sqrt {-a+\sqrt {1+a^2}}} \] Output:
-1/4*arctan(((-a+(a^2+1)^(1/2))^(1/2)-x*2^(1/2))/(a+(a^2+1)^(1/2))^(1/2))* 2^(1/2)/(a^2+1)^(1/2)/(a+(a^2+1)^(1/2))^(1/2)+1/4*arctan(((-a+(a^2+1)^(1/2 ))^(1/2)+x*2^(1/2))/(a+(a^2+1)^(1/2))^(1/2))*2^(1/2)/(a^2+1)^(1/2)/(a+(a^2 +1)^(1/2))^(1/2)+1/4*arctanh(2^(1/2)*(-a+(a^2+1)^(1/2))^(1/2)*x/((a^2+1)^( 1/2)+x^2))*2^(1/2)/(a^2+1)^(1/2)/(-a+(a^2+1)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.23 \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=-\frac {1}{2} i \left (\frac {\arctan \left (\frac {x}{\sqrt {-i+a}}\right )}{\sqrt {-i+a}}-\frac {\arctan \left (\frac {x}{\sqrt {i+a}}\right )}{\sqrt {i+a}}\right ) \] Input:
Integrate[(1 + a^2 + 2*a*x^2 + x^4)^(-1),x]
Output:
(-1/2*I)*(ArcTan[x/Sqrt[-I + a]]/Sqrt[-I + a] - ArcTan[x/Sqrt[I + a]]/Sqrt [I + a])
Time = 0.92 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.42, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1407, 1142, 25, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{a^2+2 a x^2+x^4+1} \, dx\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {\int \frac {\sqrt {2} \sqrt {\sqrt {a^2+1}-a}-x}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\int \frac {x+\sqrt {2} \sqrt {\sqrt {a^2+1}-a}}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a^2+1}-a} \int \frac {1}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}-\frac {1}{2} \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x\right )}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\frac {\sqrt {\sqrt {a^2+1}-a} \int \frac {1}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {2} x+\sqrt {\sqrt {a^2+1}-a}\right )}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a^2+1}-a} \int \frac {1}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x\right )}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\frac {\sqrt {\sqrt {a^2+1}-a} \int \frac {1}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {2} x+\sqrt {\sqrt {a^2+1}-a}\right )}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a^2+1}-a} \int \frac {1}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\frac {\sqrt {\sqrt {a^2+1}-a} \int \frac {1}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {2} x+\sqrt {\sqrt {a^2+1}-a}}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} \int \frac {1}{-\left (2 x-\sqrt {2} \sqrt {\sqrt {a^2+1}-a}\right )^2-2 \left (a+\sqrt {a^2+1}\right )}d\left (2 x-\sqrt {2} \sqrt {\sqrt {a^2+1}-a}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\frac {\int \frac {\sqrt {2} x+\sqrt {\sqrt {a^2+1}-a}}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} \int \frac {1}{-\left (2 x+\sqrt {2} \sqrt {\sqrt {a^2+1}-a}\right )^2-2 \left (a+\sqrt {a^2+1}\right )}d\left (2 x+\sqrt {2} \sqrt {\sqrt {a^2+1}-a}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sqrt {a^2+1}-a}-\sqrt {2} x}{x^2-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {\sqrt {\sqrt {a^2+1}-a} \arctan \left (\frac {2 x-\sqrt {2} \sqrt {\sqrt {a^2+1}-a}}{\sqrt {2} \sqrt {\sqrt {a^2+1}+a}}\right )}{\sqrt {\sqrt {a^2+1}+a}}}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\frac {\int \frac {\sqrt {2} x+\sqrt {\sqrt {a^2+1}-a}}{x^2+\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}}dx}{\sqrt {2}}+\frac {\sqrt {\sqrt {a^2+1}-a} \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+1}-a}+2 x}{\sqrt {2} \sqrt {\sqrt {a^2+1}+a}}\right )}{\sqrt {\sqrt {a^2+1}+a}}}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a^2+1}-a} \arctan \left (\frac {2 x-\sqrt {2} \sqrt {\sqrt {a^2+1}-a}}{\sqrt {2} \sqrt {\sqrt {a^2+1}+a}}\right )}{\sqrt {\sqrt {a^2+1}+a}}-\frac {1}{2} \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}+x^2\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}+\frac {\frac {\sqrt {\sqrt {a^2+1}-a} \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+1}-a}+2 x}{\sqrt {2} \sqrt {\sqrt {a^2+1}+a}}\right )}{\sqrt {\sqrt {a^2+1}+a}}+\frac {1}{2} \log \left (\sqrt {2} \sqrt {\sqrt {a^2+1}-a} x+\sqrt {a^2+1}+x^2\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {\sqrt {a^2+1}-a}}\) |
Input:
Int[(1 + a^2 + 2*a*x^2 + x^4)^(-1),x]
Output:
((Sqrt[-a + Sqrt[1 + a^2]]*ArcTan[(-(Sqrt[2]*Sqrt[-a + Sqrt[1 + a^2]]) + 2 *x)/(Sqrt[2]*Sqrt[a + Sqrt[1 + a^2]])])/Sqrt[a + Sqrt[1 + a^2]] - Log[Sqrt [1 + a^2] - Sqrt[2]*Sqrt[-a + Sqrt[1 + a^2]]*x + x^2]/2)/(2*Sqrt[2]*Sqrt[1 + a^2]*Sqrt[-a + Sqrt[1 + a^2]]) + ((Sqrt[-a + Sqrt[1 + a^2]]*ArcTan[(Sqr t[2]*Sqrt[-a + Sqrt[1 + a^2]] + 2*x)/(Sqrt[2]*Sqrt[a + Sqrt[1 + a^2]])])/S qrt[a + Sqrt[1 + a^2]] + Log[Sqrt[1 + a^2] + Sqrt[2]*Sqrt[-a + Sqrt[1 + a^ 2]]*x + x^2]/2)/(2*Sqrt[2]*Sqrt[1 + a^2]*Sqrt[-a + Sqrt[1 + a^2]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.16
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a^{2}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R} a}\right )}{4}\) | \(37\) |
| default | \(\frac {\frac {\left (-\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{2}-\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{3}-\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}-\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a \right ) \ln \left (x^{2}-x \sqrt {2 \sqrt {a^{2}+1}-2 a}+\sqrt {a^{2}+1}\right )}{2}+\frac {2 \left (2 a^{2}+2+\frac {\left (-\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{2}-\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{3}-\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}-\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a \right ) \sqrt {2 \sqrt {a^{2}+1}-2 a}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {2 \sqrt {a^{2}+1}-2 a}}{\sqrt {2 \sqrt {a^{2}+1}+2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+1}+2 a}}}{4 \left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {\frac {\left (\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{3}+\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}+\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a \right ) \ln \left (x^{2}+x \sqrt {2 \sqrt {a^{2}+1}-2 a}+\sqrt {a^{2}+1}\right )}{2}+\frac {2 \left (2 a^{2}+2-\frac {\left (\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a^{3}+\sqrt {a^{2}+1}\, \sqrt {2 \sqrt {a^{2}+1}-2 a}+\sqrt {2 \sqrt {a^{2}+1}-2 a}\, a \right ) \sqrt {2 \sqrt {a^{2}+1}-2 a}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {2 \sqrt {a^{2}+1}-2 a}}{\sqrt {2 \sqrt {a^{2}+1}+2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+1}+2 a}}}{4 \left (a^{2}+1\right )^{\frac {3}{2}}}\) | \(593\) |
Input:
int(1/(x^4+2*a*x^2+a^2+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(1/(_R^3+_R*a)*ln(x-_R),_R=RootOf(_Z^4+2*_Z^2*a+a^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (171) = 342\).
Time = 0.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.79 \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} + a}{a^{2} + 1}} \log \left ({\left ({\left (a^{3} + a\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} + 1\right )} \sqrt {\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} + a}{a^{2} + 1}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} + a}{a^{2} + 1}} \log \left (-{\left ({\left (a^{3} + a\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} + 1\right )} \sqrt {\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} + a}{a^{2} + 1}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} - a}{a^{2} + 1}} \log \left ({\left ({\left (a^{3} + a\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} - 1\right )} \sqrt {-\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} - a}{a^{2} + 1}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} - a}{a^{2} + 1}} \log \left (-{\left ({\left (a^{3} + a\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} - 1\right )} \sqrt {-\frac {{\left (a^{2} + 1\right )} \sqrt {-\frac {1}{a^{4} + 2 \, a^{2} + 1}} - a}{a^{2} + 1}} + x\right ) \] Input:
integrate(1/(x^4+2*a*x^2+a^2+1),x, algorithm="fricas")
Output:
1/4*sqrt(((a^2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) + a)/(a^2 + 1))*log(((a^3 + a)*sqrt(-1/(a^4 + 2*a^2 + 1)) + 1)*sqrt(((a^2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) + a)/(a^2 + 1)) + x) - 1/4*sqrt(((a^2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) + a)/(a^2 + 1))*log(-((a^3 + a)*sqrt(-1/(a^4 + 2*a^2 + 1)) + 1)*sqrt(((a^ 2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) + a)/(a^2 + 1)) + x) - 1/4*sqrt(-((a^2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) - a)/(a^2 + 1))*log(((a^3 + a)*sqrt(-1/(a^4 + 2*a^2 + 1)) - 1)*sqrt(-((a^2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) - a)/(a^2 + 1)) + x) + 1/4*sqrt(-((a^2 + 1)*sqrt(-1/(a^4 + 2*a^2 + 1)) - a)/(a^2 + 1 ))*log(-((a^3 + a)*sqrt(-1/(a^4 + 2*a^2 + 1)) - 1)*sqrt(-((a^2 + 1)*sqrt(- 1/(a^4 + 2*a^2 + 1)) - a)/(a^2 + 1)) + x)
Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.21 \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} + 256\right ) - 32 t^{2} a + 1, \left ( t \mapsto t \log {\left (64 t^{3} a^{3} + 64 t^{3} a - 4 t a^{2} + 4 t + x \right )} \right )\right )} \] Input:
integrate(1/(x**4+2*a*x**2+a**2+1),x)
Output:
RootSum(_t**4*(256*a**2 + 256) - 32*_t**2*a + 1, Lambda(_t, _t*log(64*_t** 3*a**3 + 64*_t**3*a - 4*_t*a**2 + 4*_t + x)))
\[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=\int { \frac {1}{x^{4} + 2 \, a x^{2} + a^{2} + 1} \,d x } \] Input:
integrate(1/(x^4+2*a*x^2+a^2+1),x, algorithm="maxima")
Output:
integrate(1/(x^4 + 2*a*x^2 + a^2 + 1), x)
Timed out. \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=\text {Timed out} \] Input:
integrate(1/(x^4+2*a*x^2+a^2+1),x, algorithm="giac")
Output:
Timed out
Time = 17.38 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.07 \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=-\frac {\mathrm {atanh}\left (-\frac {2\,x\,\sqrt {\frac {a}{a^2+1}+\frac {1{}\mathrm {i}}{a^2+1}}}{\frac {2\,a}{a^2+1}+\frac {2{}\mathrm {i}}{a^2+1}}+\frac {a\,x\,\sqrt {\frac {a}{a^2+1}+\frac {1{}\mathrm {i}}{a^2+1}}\,2{}\mathrm {i}}{\frac {2\,a}{a^2+1}+\frac {2\,a^3}{a^2+1}+\frac {2{}\mathrm {i}}{a^2+1}+\frac {a^2\,2{}\mathrm {i}}{a^2+1}}+\frac {2\,a^2\,x\,\sqrt {\frac {a}{a^2+1}+\frac {1{}\mathrm {i}}{a^2+1}}}{\frac {2\,a}{a^2+1}+\frac {2\,a^3}{a^2+1}+\frac {2{}\mathrm {i}}{a^2+1}+\frac {a^2\,2{}\mathrm {i}}{a^2+1}}\right )\,\sqrt {\frac {a+1{}\mathrm {i}}{a^2+1}}}{2}+2\,\mathrm {atanh}\left (\frac {8\,x\,\sqrt {\frac {a}{16\,a^2+16}-\frac {1{}\mathrm {i}}{16\,a^2+16}}}{\frac {32\,a}{16\,a^2+16}-\frac {32{}\mathrm {i}}{16\,a^2+16}}+\frac {a\,x\,\sqrt {\frac {a}{16\,a^2+16}-\frac {1{}\mathrm {i}}{16\,a^2+16}}\,128{}\mathrm {i}}{\frac {512\,a}{16\,a^2+16}+\frac {512\,a^3}{16\,a^2+16}-\frac {512{}\mathrm {i}}{16\,a^2+16}-\frac {a^2\,512{}\mathrm {i}}{16\,a^2+16}}-\frac {128\,a^2\,x\,\sqrt {\frac {a}{16\,a^2+16}-\frac {1{}\mathrm {i}}{16\,a^2+16}}}{\frac {512\,a}{16\,a^2+16}+\frac {512\,a^3}{16\,a^2+16}-\frac {512{}\mathrm {i}}{16\,a^2+16}-\frac {a^2\,512{}\mathrm {i}}{16\,a^2+16}}\right )\,\sqrt {\frac {a-\mathrm {i}}{16\,a^2+16}} \] Input:
int(1/(2*a*x^2 + a^2 + x^4 + 1),x)
Output:
2*atanh((8*x*(a/(16*a^2 + 16) - 1i/(16*a^2 + 16))^(1/2))/((32*a)/(16*a^2 + 16) - 32i/(16*a^2 + 16)) + (a*x*(a/(16*a^2 + 16) - 1i/(16*a^2 + 16))^(1/2 )*128i)/((512*a)/(16*a^2 + 16) - 512i/(16*a^2 + 16) - (a^2*512i)/(16*a^2 + 16) + (512*a^3)/(16*a^2 + 16)) - (128*a^2*x*(a/(16*a^2 + 16) - 1i/(16*a^2 + 16))^(1/2))/((512*a)/(16*a^2 + 16) - 512i/(16*a^2 + 16) - (a^2*512i)/(1 6*a^2 + 16) + (512*a^3)/(16*a^2 + 16)))*((a - 1i)/(16*a^2 + 16))^(1/2) - ( atanh((a*x*(a/(a^2 + 1) + 1i/(a^2 + 1))^(1/2)*2i)/((2*a)/(a^2 + 1) + 2i/(a ^2 + 1) + (a^2*2i)/(a^2 + 1) + (2*a^3)/(a^2 + 1)) - (2*x*(a/(a^2 + 1) + 1i /(a^2 + 1))^(1/2))/((2*a)/(a^2 + 1) + 2i/(a^2 + 1)) + (2*a^2*x*(a/(a^2 + 1 ) + 1i/(a^2 + 1))^(1/2))/((2*a)/(a^2 + 1) + 2i/(a^2 + 1) + (a^2*2i)/(a^2 + 1) + (2*a^3)/(a^2 + 1)))*((a + 1i)/(a^2 + 1))^(1/2))/2
Time = 0.22 (sec) , antiderivative size = 563, normalized size of antiderivative = 2.48 \[ \int \frac {1}{1+a^2+2 a x^2+x^4} \, dx=\frac {\sqrt {2}\, \left (2 \sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}-2 x}{\sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {2}}\right ) a -2 \sqrt {\sqrt {a^{2}+1}+a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}-2 x}{\sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {2}}\right ) a^{2}-2 \sqrt {\sqrt {a^{2}+1}+a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}-2 x}{\sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {2}}\right )-2 \sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}+2 x}{\sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {2}}\right ) a +2 \sqrt {\sqrt {a^{2}+1}+a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}+2 x}{\sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {2}}\right ) a^{2}+2 \sqrt {\sqrt {a^{2}+1}+a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}+2 x}{\sqrt {\sqrt {a^{2}+1}+a}\, \sqrt {2}}\right )-\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {a^{2}+1}\, \mathrm {log}\left (\sqrt {a^{2}+1}-\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}\, x +x^{2}\right ) a +\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {a^{2}+1}\, \mathrm {log}\left (\sqrt {a^{2}+1}+\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}\, x +x^{2}\right ) a -\sqrt {\sqrt {a^{2}+1}-a}\, \mathrm {log}\left (\sqrt {a^{2}+1}-\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}\, x +x^{2}\right ) a^{2}-\sqrt {\sqrt {a^{2}+1}-a}\, \mathrm {log}\left (\sqrt {a^{2}+1}-\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}\, x +x^{2}\right )+\sqrt {\sqrt {a^{2}+1}-a}\, \mathrm {log}\left (\sqrt {a^{2}+1}+\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}\, x +x^{2}\right ) a^{2}+\sqrt {\sqrt {a^{2}+1}-a}\, \mathrm {log}\left (\sqrt {a^{2}+1}+\sqrt {\sqrt {a^{2}+1}-a}\, \sqrt {2}\, x +x^{2}\right )\right )}{8 a^{2}+8} \] Input:
int(1/(x^4+2*a*x^2+a^2+1),x)
Output:
(sqrt(2)*(2*sqrt(sqrt(a**2 + 1) + a)*sqrt(a**2 + 1)*atan((sqrt(sqrt(a**2 + 1) - a)*sqrt(2) - 2*x)/(sqrt(sqrt(a**2 + 1) + a)*sqrt(2)))*a - 2*sqrt(sqr t(a**2 + 1) + a)*atan((sqrt(sqrt(a**2 + 1) - a)*sqrt(2) - 2*x)/(sqrt(sqrt( a**2 + 1) + a)*sqrt(2)))*a**2 - 2*sqrt(sqrt(a**2 + 1) + a)*atan((sqrt(sqrt (a**2 + 1) - a)*sqrt(2) - 2*x)/(sqrt(sqrt(a**2 + 1) + a)*sqrt(2))) - 2*sqr t(sqrt(a**2 + 1) + a)*sqrt(a**2 + 1)*atan((sqrt(sqrt(a**2 + 1) - a)*sqrt(2 ) + 2*x)/(sqrt(sqrt(a**2 + 1) + a)*sqrt(2)))*a + 2*sqrt(sqrt(a**2 + 1) + a )*atan((sqrt(sqrt(a**2 + 1) - a)*sqrt(2) + 2*x)/(sqrt(sqrt(a**2 + 1) + a)* sqrt(2)))*a**2 + 2*sqrt(sqrt(a**2 + 1) + a)*atan((sqrt(sqrt(a**2 + 1) - a) *sqrt(2) + 2*x)/(sqrt(sqrt(a**2 + 1) + a)*sqrt(2))) - sqrt(sqrt(a**2 + 1) - a)*sqrt(a**2 + 1)*log(sqrt(a**2 + 1) - sqrt(sqrt(a**2 + 1) - a)*sqrt(2)* x + x**2)*a + sqrt(sqrt(a**2 + 1) - a)*sqrt(a**2 + 1)*log(sqrt(a**2 + 1) + sqrt(sqrt(a**2 + 1) - a)*sqrt(2)*x + x**2)*a - sqrt(sqrt(a**2 + 1) - a)*l og(sqrt(a**2 + 1) - sqrt(sqrt(a**2 + 1) - a)*sqrt(2)*x + x**2)*a**2 - sqrt (sqrt(a**2 + 1) - a)*log(sqrt(a**2 + 1) - sqrt(sqrt(a**2 + 1) - a)*sqrt(2) *x + x**2) + sqrt(sqrt(a**2 + 1) - a)*log(sqrt(a**2 + 1) + sqrt(sqrt(a**2 + 1) - a)*sqrt(2)*x + x**2)*a**2 + sqrt(sqrt(a**2 + 1) - a)*log(sqrt(a**2 + 1) + sqrt(sqrt(a**2 + 1) - a)*sqrt(2)*x + x**2)))/(8*(a**2 + 1))