Integrand size = 18, antiderivative size = 109 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt {b}} \] Output:
1/2*arctan(a^(1/4)*x/(a^(1/2)-b^(1/2))^(1/2))/a^(1/4)/(a^(1/2)-b^(1/2))^(1 /2)/b^(1/2)-1/2*arctan(a^(1/4)*x/(a^(1/2)+b^(1/2))^(1/2))/a^(1/4)/(a^(1/2) +b^(1/2))^(1/2)/b^(1/2)
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a-\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:
Integrate[(a - b + 2*a*x^2 + a*x^4)^(-1),x]
Output:
ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a - Sqrt[a]*Sqrt[b]] *Sqrt[b]) - ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a + Sqrt [a]*Sqrt[b]]*Sqrt[b])
Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1406, 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.
\(\displaystyle \int \frac {1}{a x^4+2 a x^2+a-b} \, dx\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {\sqrt {a} \int \frac {1}{a x^2+\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}dx}{2 \sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{a x^2+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}dx}{2 \sqrt {b}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {b} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {b} \sqrt {\sqrt {a}+\sqrt {b}}}\) |
Input:
Int[(a - b + 2*a*x^2 + a*x^4)^(-1),x]
Output:
ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt [b]]*Sqrt[b]) - ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*a^(1/4)*Sqr t[Sqrt[a] + Sqrt[b]]*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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a}\) | \(40\) |
default | \(a \left (-\frac {\operatorname {arctanh}\left (\frac {a x}{\sqrt {\left (\sqrt {a b}-a \right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) a}}-\frac {\arctan \left (\frac {a x}{\sqrt {\left (\sqrt {a b}+a \right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) a}}\right )\) | \(74\) |
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. 553 vs. \(2 (69) = 138\).
Time = 0.08 (sec) , antiderivative size = 553, normalized size of antiderivative = 5.07 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=-\frac {1}{4} \, \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} \log \left ({\left (b - \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} \log \left (-{\left (b - \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} \log \left ({\left (b + \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} \log \left (-{\left (b + \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} + x\right ) \] Input:
integrate(1/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
Output:
-1/4*sqrt(-((a*b - b^2)/sqrt(a^3*b - 2*a^2*b^2 + a*b^3) + 1)/(a*b - b^2))* log((b - (a^2*b - a*b^2)/sqrt(a^3*b - 2*a^2*b^2 + a*b^3))*sqrt(-((a*b - b^ 2)/sqrt(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(a^3*b - 2*a^2*b^2 + a*b^3) + 1)/(a*b - b^2))*log(-(b - (a^2* b - a*b^2)/sqrt(a^3*b - 2*a^2*b^2 + a*b^3))*sqrt(-((a*b - b^2)/sqrt(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(a ^3*b - 2*a^2*b^2 + a*b^3) - 1)/(a*b - b^2))*log((b + (a^2*b - a*b^2)/sqrt( a^3*b - 2*a^2*b^2 + a*b^3))*sqrt(((a*b - b^2)/sqrt(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(a^3*b - 2*a^2*b^2 + a*b^3) - 1)/(a*b - b^2))*log(-(b + (a^2*b - a*b^2)/sqrt(a^3*b - 2*a^2*b^ 2 + a*b^3))*sqrt(((a*b - b^2)/sqrt(a^3*b - 2*a^2*b^2 + a*b^3) - 1)/(a*b - b^2)) + x)
Time = 0.44 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} b^{2} - 256 a b^{3}\right ) + 32 t^{2} a b + 1, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} b + 64 t^{3} a b^{2} - 4 t a - 4 t b + x \right )} \right )\right )} \] 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)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (69) = 138\).
Time = 0.17 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.74 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{2} + 3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{5} b - 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{2} + 3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{5} b - 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} \] 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)*sqrt(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)*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)
Time = 20.37 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.95 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\ln \left (4\,a^3\,b\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}-4\,a^3\,x+\frac {4\,a^4\,b\,x}{a\,b+\sqrt {a\,b^3}}\right )\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}}{4}+\frac {\ln \left (4\,a^3\,x-4\,a^3\,b\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}-\frac {4\,a^4\,b\,x}{a\,b-\sqrt {a\,b^3}}\right )\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}}{4}-\ln \left (4\,a^3\,x+4\,a^3\,b\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}-\frac {4\,a^4\,b\,x}{a\,b+\sqrt {a\,b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {a\,b^3}}{16\,\left (a\,b^3-a^2\,b^2\right )}}-\ln \left (4\,a^3\,x+16\,a^3\,b\,\sqrt {-\frac {1}{16\,a\,b-16\,\sqrt {a\,b^3}}}-\frac {4\,a^4\,b\,x}{a\,b-\sqrt {a\,b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {a\,b^3}}{16\,\left (a\,b^3-a^2\,b^2\right )}} \] Input:
int(1/(a - b + 2*a*x^2 + a*x^4),x)
Output:
(log(4*a^3*b*(-1/(a*b + (a*b^3)^(1/2)))^(1/2) - 4*a^3*x + (4*a^4*b*x)/(a*b + (a*b^3)^(1/2)))*(-1/(a*b + (a*b^3)^(1/2)))^(1/2))/4 + (log(4*a^3*x - 4* a^3*b*(-1/(a*b - (a*b^3)^(1/2)))^(1/2) - (4*a^4*b*x)/(a*b - (a*b^3)^(1/2)) )*(-1/(a*b - (a*b^3)^(1/2)))^(1/2))/4 - log(4*a^3*x + 4*a^3*b*(-1/(a*b + ( a*b^3)^(1/2)))^(1/2) - (4*a^4*b*x)/(a*b + (a*b^3)^(1/2)))*((a*b - (a*b^3)^ (1/2))/(16*(a*b^3 - a^2*b^2)))^(1/2) - log(4*a^3*x + 16*a^3*b*(-1/(16*a*b - 16*(a*b^3)^(1/2)))^(1/2) - (4*a^4*b*x)/(a*b - (a*b^3)^(1/2)))*((a*b + (a *b^3)^(1/2))/(16*(a*b^3 - a^2*b^2)))^(1/2)
Time = 0.15 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.87 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}\, \mathit {atan} \left (\frac {a x}{\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}}\right ) b -2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}\, \mathit {atan} \left (\frac {a x}{\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}+a}}\right ) a -\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b -\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}-a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a}{4 a b \left (a -b \right )} \] Input:
int(1/(a*x^4+2*a*x^2+a-b),x)
Output:
(2*sqrt(a)*sqrt(sqrt(b)*sqrt(a) + a)*atan((a*x)/(sqrt(a)*sqrt(sqrt(b)*sqrt (a) + a)))*b - 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a) + a)*atan((a*x)/(sqrt(a)*sqr t(sqrt(b)*sqrt(a) + a)))*a - sqrt(a)*sqrt(sqrt(b)*sqrt(a) - a)*log( - sqrt (sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b + sqrt(a)*sqrt(sqrt(b)*sqrt(a) - a)*l og(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b - sqrt(b)*sqrt(sqrt(b)*sqrt(a) - a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a + sqrt(b)*sqrt(sqrt( b)*sqrt(a) - a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a)/(4*a*b*(a - b))