Integrand size = 22, antiderivative size = 109 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=-\frac {\sqrt {\sqrt {a}-\sqrt {b}} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\sqrt {\sqrt {a}+\sqrt {b}} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {b}} \] Output:
-1/2*(a^(1/2)-b^(1/2))^(1/2)*arctan(a^(1/4)*x/(a^(1/2)-b^(1/2))^(1/2))/a^( 3/4)/b^(1/2)+1/2*(a^(1/2)+b^(1/2))^(1/2)*arctan(a^(1/4)*x/(a^(1/2)+b^(1/2) )^(1/2))/a^(3/4)/b^(1/2)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=\frac {-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a-\sqrt {a} \sqrt {b}}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}}{2 \sqrt {a} \sqrt {b}} \] Input:
Integrate[x^2/(a - b + 2*a*x^2 + a*x^4),x]
Output:
(-(((Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]])/Sqr t[a - Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt[b])
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1450, 218}
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.
\(\displaystyle \int \frac {x^2}{a x^4+2 a x^2+a-b} \, dx\) |
\(\Big \downarrow \) 1450 |
\(\displaystyle \frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {b}}\right ) \int \frac {1}{a x^2+\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}dx+\frac {1}{2} \left (\frac {\sqrt {a}}{\sqrt {b}}+1\right ) \int \frac {1}{a x^2+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}dx\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\left (1-\frac {\sqrt {a}}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (\frac {\sqrt {a}}{\sqrt {b}}+1\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}\) |
Input:
Int[x^2/(a - b + 2*a*x^2 + a*x^4),x]
Output:
((1 - Sqrt[a]/Sqrt[b])*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^( 3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + ((1 + Sqrt[a]/Sqrt[b])*ArcTan[(a^(1/4)*x)/ Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]])
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[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1) Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.39
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a}\) | \(43\) |
default | \(a \left (-\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {a x}{\sqrt {\left (\sqrt {a b}-a \right ) a}}\right )}{2 a \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) a}}+\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {a x}{\sqrt {\left (\sqrt {a b}+a \right ) a}}\right )}{2 a \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) a}}\right )\) | \(96\) |
Input:
int(x^2/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)
Output:
1/4/a*sum(_R^2/(_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. 267 vs. \(2 (69) = 138\).
Time = 0.08 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.45 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=\frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) + \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) \] Input:
integrate(x^2/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
Output:
1/4*sqrt(-(a*b*sqrt(1/(a^3*b)) + 1)/(a*b))*log(a^2*b*sqrt(-(a*b*sqrt(1/(a^ 3*b)) + 1)/(a*b))*sqrt(1/(a^3*b)) + x) - 1/4*sqrt(-(a*b*sqrt(1/(a^3*b)) + 1)/(a*b))*log(-a^2*b*sqrt(-(a*b*sqrt(1/(a^3*b)) + 1)/(a*b))*sqrt(1/(a^3*b) ) + x) - 1/4*sqrt((a*b*sqrt(1/(a^3*b)) - 1)/(a*b))*log(a^2*b*sqrt((a*b*sqr t(1/(a^3*b)) - 1)/(a*b))*sqrt(1/(a^3*b)) + x) + 1/4*sqrt((a*b*sqrt(1/(a^3* b)) - 1)/(a*b))*log(-a^2*b*sqrt((a*b*sqrt(1/(a^3*b)) - 1)/(a*b))*sqrt(1/(a ^3*b)) + x)
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.40 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} + 32 t^{2} a^{2} b + a - b, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \] Input:
integrate(x**2/(a*x**4+2*a*x**2+a-b),x)
Output:
RootSum(256*_t**4*a**3*b**2 + 32*_t**2*a**2*b + a - b, Lambda(_t, _t*log(- 64*_t**3*a**2*b - 4*_t*a + x)))
\[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=\int { \frac {x^{2}}{a x^{4} + 2 \, a x^{2} + a - b} \,d x } \] Input:
integrate(x^2/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")
Output:
integrate(x^2/(a*x^4 + 2*a*x^2 + a - b), x)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (69) = 138\).
Time = 0.23 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.83 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b - 4 \, a^{3} b^{2}\right )}} - \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b - 4 \, a^{3} b^{2}\right )}} \] Input:
integrate(x^2/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")
Output:
1/2*(3*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a - 4*sqrt(a^2 + sqrt(a*b)*a)*sqr t(a*b)*b)*abs(a)*arctan(2*sqrt(1/2)*x/sqrt((2*a + sqrt(-4*(a - b)*a + 4*a^ 2))/a))/(3*a^4*b - 4*a^3*b^2) - 1/2*(3*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a - 4*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*b)*abs(a)*arctan(2*sqrt(1/2)*x/sqrt ((2*a - sqrt(-4*(a - b)*a + 4*a^2))/a))/(3*a^4*b - 4*a^3*b^2)
Time = 0.30 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.98 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^3+4\,b\,a^2\right )-\frac {4\,a\,x\,\left (\sqrt {a^3\,b^3}+a^2\,b\right )}{b}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}}{2\,a\,b-2\,a^2}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}-2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^3+4\,b\,a^2\right )+\frac {4\,a\,x\,\left (\sqrt {a^3\,b^3}-a^2\,b\right )}{b}\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}}}{2\,a\,b-2\,a^2}\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}} \] Input:
int(x^2/(a - b + 2*a*x^2 + a*x^4),x)
Output:
- 2*atanh((2*(x*(4*a^2*b + 4*a^3) - (4*a*x*((a^3*b^3)^(1/2) + a^2*b))/b)*( -((a^3*b^3)^(1/2) + a^2*b)/(16*a^3*b^2))^(1/2))/(2*a*b - 2*a^2))*(-((a^3*b ^3)^(1/2) + a^2*b)/(16*a^3*b^2))^(1/2) - 2*atanh((2*(x*(4*a^2*b + 4*a^3) + (4*a*x*((a^3*b^3)^(1/2) - a^2*b))/b)*(((a^3*b^3)^(1/2) - a^2*b)/(16*a^3*b ^2))^(1/2))/(2*a*b - 2*a^2))*(((a^3*b^3)^(1/2) - a^2*b)/(16*a^3*b^2))^(1/2 )
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{a-b+2 a x^2+a x^4} \, dx=\frac {\sqrt {b}\, \left (2 \sqrt {\sqrt {b}\, \sqrt {a}+a}\, \mathit {atan} \left (\frac {a x}{\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}}\right )+\sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right )-\sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right )\right )}{4 a b} \] Input:
int(x^2/(a*x^4+2*a*x^2+a-b),x)
Output:
(sqrt(b)*(2*sqrt(sqrt(b)*sqrt(a) + a)*atan((a*x)/(sqrt(a)*sqrt(sqrt(b)*sqr t(a) + a))) + sqrt(sqrt(b)*sqrt(a) - a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x) - sqrt(sqrt(b)*sqrt(a) - a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sq rt(a)*x)))/(4*a*b)