Integrand size = 16, antiderivative size = 271 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt {a+b}+\sqrt {a} x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {-\sqrt {a}+\sqrt {a+b}}} \] Output:
-1/4*arctan(((-a^(1/2)+(a+b)^(1/2))^(1/2)-2^(1/2)*a^(1/4)*x)/(a^(1/2)+(a+b )^(1/2))^(1/2))*2^(1/2)/a^(1/4)/(a+b)^(1/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)+1/ 4*arctan(((-a^(1/2)+(a+b)^(1/2))^(1/2)+2^(1/2)*a^(1/4)*x)/(a^(1/2)+(a+b)^( 1/2))^(1/2))*2^(1/2)/a^(1/4)/(a+b)^(1/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)+1/4*a rctanh(2^(1/2)*a^(1/4)*(-a^(1/2)+(a+b)^(1/2))^(1/2)*x/((a+b)^(1/2)+a^(1/2) *x^2))*2^(1/2)/a^(1/4)/(a+b)^(1/2)/(-a^(1/2)+(a+b)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.44 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=-\frac {i \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a-i \sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {i \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a+i \sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:
Integrate[(a + b + 2*a*x^2 + a*x^4)^(-1),x]
Output:
((-1/2*I)*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/(Sqrt[a - I*Sqr t[a]*Sqrt[b]]*Sqrt[b]) + ((I/2)*ArcTan[(Sqrt[a]*x)/Sqrt[a + I*Sqrt[a]*Sqrt [b]]])/(Sqrt[a + I*Sqrt[a]*Sqrt[b]]*Sqrt[b])
Time = 1.09 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.52, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1407, 27, 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.
| \(\displaystyle \int \frac {1}{a x^4+2 a x^2+a+b} \, dx\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {\int \frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} x}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt [4]{a} x+\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\int \frac {\sqrt [4]{a} x+\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{a} \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{a} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{a} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{a} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{-\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{-\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2}}+\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {1}{2} \sqrt [4]{a} \log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}+\frac {\frac {\sqrt [4]{a} \sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {1}{2} \sqrt [4]{a} \log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a+b} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
Input:
Int[(a + b + 2*a*x^2 + a*x^4)^(-1),x]
Output:
((a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*ArcTan[(a^(1/4)*(-((Sqrt[2]*Sqrt[-S qrt[a] + Sqrt[a + b]])/a^(1/4)) + 2*x))/(Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[a + b ]])])/Sqrt[Sqrt[a] + Sqrt[a + b]] - (a^(1/4)*Log[Sqrt[a + b] - Sqrt[2]*a^( 1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2])/2)/(2*Sqrt[2]*Sqrt[a]* Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) + ((a^(1/4)*Sqrt[-Sqrt[a] + Sqrt [a + b]]*ArcTan[(a^(1/4)*((Sqrt[2]*Sqrt[-Sqrt[a] + Sqrt[a + b]])/a^(1/4) + 2*x))/(Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[a + b]])])/Sqrt[Sqrt[a] + Sqrt[a + b]] + (a^(1/4)*Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]] *x + Sqrt[a]*x^2])/2)/(2*Sqrt[2]*Sqrt[a]*Sqrt[a + b]*Sqrt[-Sqrt[a] + Sqrt[ a + b]])
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 = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.14
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a}\) | \(38\) |
| default | \(\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \ln \left (\sqrt {a}\, x^{2}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (2 \sqrt {a}\, b -\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 \sqrt {a +b}\, \sqrt {a}\, b}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \ln \left (-\sqrt {a}\, x^{2}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}-\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (-2 \sqrt {a}\, b +\frac {\left (\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 \sqrt {a +b}\, \sqrt {a}\, b}\) | \(499\) |
Input:
int(1/(a*x^4+2*a*x^2+a+b),x,method=_RETURNVERBOSE)
Output:
1/4/a*sum(1/(_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4*a+2*_Z^2*a+a+b))
Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (185) = 370\).
Time = 0.08 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.09 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} \log \left ({\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + b\right )} \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} \log \left (-{\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + b\right )} \sqrt {\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} \log \left ({\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - b\right )} \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} \log \left (-{\left ({\left (a^{2} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - b\right )} \sqrt {-\frac {{\left (a b + b^{2}\right )} \sqrt {-\frac {1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} + x\right ) \] Input:
integrate(1/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")
Output:
1/4*sqrt(((a*b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) + 1)/(a*b + b^2 ))*log(((a^2*b + a*b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) + b)*sqrt(((a *b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) + 1)/(a*b + b^2)) + x) - 1/ 4*sqrt(((a*b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) + 1)/(a*b + b^2)) *log(-((a^2*b + a*b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) + b)*sqrt(((a* b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) + 1)/(a*b + b^2)) + x) - 1/4 *sqrt(-((a*b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) - 1)/(a*b + b^2)) *log(((a^2*b + a*b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) - b)*sqrt(-((a* b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) - 1)/(a*b + b^2)) + x) + 1/4 *sqrt(-((a*b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) - 1)/(a*b + b^2)) *log(-((a^2*b + a*b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) - b)*sqrt(-((a *b + b^2)*sqrt(-1/(a^3*b + 2*a^2*b^2 + a*b^3)) - 1)/(a*b + b^2)) + x)
Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.23 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} b^{2} + 256 a b^{3}\right ) - 32 t^{2} a b + 1, \left ( t \mapsto t \log {\left (64 t^{3} a^{2} b + 64 t^{3} a b^{2} - 4 t a + 4 t b + x \right )} \right )\right )} \] Input:
integrate(1/(a*x**4+2*a*x**2+a+b),x)
Output:
RootSum(_t**4*(256*a**2*b**2 + 256*a*b**3) - 32*_t**2*a*b + 1, Lambda(_t, _t*log(64*_t**3*a**2*b + 64*_t**3*a*b**2 - 4*_t*a + 4*_t*b + x)))
\[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=\int { \frac {1}{a x^{4} + 2 \, a x^{2} + a + b} \,d x } \] Input:
integrate(1/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")
Output:
integrate(1/(a*x^4 + 2*a*x^2 + a + b), x)
Time = 0.16 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.13 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} b + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b^{2} + 3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{5} b + 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} b + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b^{2} + 3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{5} b + 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} \] Input:
integrate(1/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")
Output:
1/2*(3*sqrt(a^2 + sqrt(-a*b)*a)*a^2*b + 4*sqrt(a^2 + sqrt(-a*b)*a)*a*b^2 + 3*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a^2 + 4*sqrt(a^2 + sqrt(-a*b)*a)*sq rt(-a*b)*a*b)*abs(a)*arctan(2*sqrt(1/2)*x/sqrt((2*a + sqrt(-4*(a + b)*a + 4*a^2))/a))/(3*a^5*b + 7*a^4*b^2 + 4*a^3*b^3) + 1/2*(3*sqrt(a^2 - sqrt(-a* b)*a)*a^2*b + 4*sqrt(a^2 - sqrt(-a*b)*a)*a*b^2 + 3*sqrt(a^2 - sqrt(-a*b)*a )*sqrt(-a*b)*a^2 + 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*a*b)*abs(a)*arcta n(2*sqrt(1/2)*x/sqrt((2*a - sqrt(-4*(a + b)*a + 4*a^2))/a))/(3*a^5*b + 7*a ^4*b^2 + 4*a^3*b^3)
Time = 18.63 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.64 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx=2\,\mathrm {atanh}\left (\frac {8\,a^3\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}-\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^4\,b^2}{a^2\,b^2+a\,b^3}-\frac {2\,a^3\,b\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}-\frac {8\,a^5\,b^2\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}-\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}-\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}-\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}+\frac {8\,a^4\,b\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}-\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}\,\sqrt {-a\,b^3}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}-\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}-\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^5\,b^2\,x\,\sqrt {\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}+\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}+\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}-\frac {8\,a^3\,x\,\sqrt {\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}}}{\frac {2\,a^4\,b^2}{a^2\,b^2+a\,b^3}+\frac {2\,a^3\,b\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}+\frac {8\,a^4\,b\,x\,\sqrt {\frac {\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+a\,b^3\right )}}\,\sqrt {-a\,b^3}}{\frac {2\,a^5\,b^5}{a^2\,b^2+a\,b^3}+\frac {2\,a^6\,b^4}{a^2\,b^2+a\,b^3}+\frac {2\,a^4\,b^4\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}+\frac {2\,a^5\,b^3\,\sqrt {-a\,b^3}}{a^2\,b^2+a\,b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {-a\,b^3}}{16\,\left (a^2\,b^2+a\,b^3\right )}} \] Input:
int(1/(a + b + 2*a*x^2 + a*x^4),x)
Output:
2*atanh((8*a^3*x*((a*b)/(16*(a*b^3 + a^2*b^2)) - (-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^4*b^2)/(a*b^3 + a^2*b^2) - (2*a^3*b*(-a*b^3)^(1 /2))/(a*b^3 + a^2*b^2)) - (8*a^5*b^2*x*((a*b)/(16*(a*b^3 + a^2*b^2)) - (-a *b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a^6*b^4)/(a*b^3 + a^2*b^2) - (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2* b^2) - (2*a^5*b^3*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2)) + (8*a^4*b*x*((a*b)/( 16*(a*b^3 + a^2*b^2)) - (-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)))^(1/2)*(-a*b ^3)^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a^6*b^4)/(a*b^3 + a^2*b^2) - (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) - (2*a^5*b^3*(-a*b^3)^(1/2) )/(a*b^3 + a^2*b^2)))*((a*b - (-a*b^3)^(1/2))/(16*(a*b^3 + a^2*b^2)))^(1/2 ) - 2*atanh((8*a^5*b^2*x*((-a*b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)) + (a*b)/(1 6*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a^6*b^4)/ (a*b^3 + a^2*b^2) + (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) + (2*a^5* b^3*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2)) - (8*a^3*x*((-a*b^3)^(1/2)/(16*(a*b ^3 + a^2*b^2)) + (a*b)/(16*(a*b^3 + a^2*b^2)))^(1/2))/((2*a^4*b^2)/(a*b^3 + a^2*b^2) + (2*a^3*b*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2)) + (8*a^4*b*x*((-a *b^3)^(1/2)/(16*(a*b^3 + a^2*b^2)) + (a*b)/(16*(a*b^3 + a^2*b^2)))^(1/2)*( -a*b^3)^(1/2))/((2*a^5*b^5)/(a*b^3 + a^2*b^2) + (2*a^6*b^4)/(a*b^3 + a^2*b ^2) + (2*a^4*b^4*(-a*b^3)^(1/2))/(a*b^3 + a^2*b^2) + (2*a^5*b^3*(-a*b^3)^( 1/2))/(a*b^3 + a^2*b^2)))*((a*b + (-a*b^3)^(1/2))/(16*(a*b^3 + a^2*b^2)...
Time = 0.15 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.28 \[ \int \frac {1}{a+b+2 a x^2+a x^4} \, dx =\text {Too large to display} \] Input:
int(1/(a*x^4+2*a*x^2+a+b),x)
Output:
(sqrt(2)*(2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) + a)*atan((sqrt(sqrt(a)*s qrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt (2)))*a - 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*atan((sqrt(sqrt(a)*sqrt( a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)) )*a - 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b - 2*sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) + a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a + 2*sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a + 2* sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) + a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)* sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b - sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) - a)*log( - sqrt(sqrt(a)*sqrt(a + b) - a)*s qrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a + sqrt(a + b)*sqrt(sqrt(a)*sqrt(a + b) - a)*log(sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqr t(a)*x**2)*a - sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) - a)*log( - sqrt(sqrt(a)*s qrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a - sqrt(a)*sqrt(s qrt(a)*sqrt(a + b) - a)*log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + s qrt(a + b) + sqrt(a)*x**2)*b + sqrt(a)*sqrt(sqrt(a)*sqrt(a + b) - a)*log(s qrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a ...