\(\int \frac {1}{a-b+2 a x^2+a x^4} \, dx\) [904]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt {b}} \]

[Out]

1/2*arctan(a^(1/4)*x/(a^(1/2)-b^(1/2))^(1/2))/a^(1/4)/b^(1/2)/(a^(1/2)-b^(1/2))^(1/2)-1/2*arctan(a^(1/4)*x/(a^
(1/2)+b^(1/2))^(1/2))/a^(1/4)/b^(1/2)/(a^(1/2)+b^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1107, 211} \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {b} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {b} \sqrt {\sqrt {a}+\sqrt {b}}} \]

[In]

Int[(a - b + 2*a*x^2 + a*x^4)^(-1),x]

[Out]

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)/S
qrt[Sqrt[a] + Sqrt[b]]]/(2*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]]*Sqrt[b])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a} \int \frac {1}{a-\sqrt {a} \sqrt {b}+a x^2} \, dx}{2 \sqrt {b}}-\frac {\sqrt {a} \int \frac {1}{a+\sqrt {a} \sqrt {b}+a x^2} \, dx}{2 \sqrt {b}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a-\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}} \]

[In]

Integrate[(a - b + 2*a*x^2 + a*x^4)^(-1),x]

[Out]

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])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.04 (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	\(a \left (-\frac {\operatorname {arctanh}\left (\frac {a x}{\sqrt {\left (\sqrt {a b}-a \right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) a}}-\frac {\arctan \left (\frac {a x}{\sqrt {\left (\sqrt {a b}+a \right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) a}}\right )\)	\(74\)

[In]

int(1/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)

[Out]

1/4/a*sum(1/(_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4*a+2*_Z^2*a+a-b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (69) = 138\).

Time = 0.24 (sec) , antiderivative size = 553, normalized size of antiderivative = 5.07 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=-\frac {1}{4} \, \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} \log \left ({\left (b - \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} \log \left (-{\left (b - \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {-\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b - b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} \log \left ({\left (b + \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} \log \left (-{\left (b + \frac {a^{2} b - a b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}\right )} \sqrt {\frac {\frac {a b - b^{2}}{\sqrt {a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b - b^{2}}} + x\right ) \]

[In]

integrate(1/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")

[Out]

-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)/sq
rt(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)

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} b^{2} - 256 a b^{3}\right ) + 32 t^{2} a b + 1, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} b + 64 t^{3} a b^{2} - 4 t a - 4 t b + x \right )} \right )\right )} \]

[In]

integrate(1/(a*x**4+2*a*x**2+a-b),x)

[Out]

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)))

Maxima [F]

\[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\int { \frac {1}{a x^{4} + 2 \, a x^{2} + a - b} \,d x } \]

[In]

integrate(1/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")

[Out]

integrate(1/(a*x^4 + 2*a*x^2 + a - b), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (69) = 138\).

Time = 0.31 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.74 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{2} + 3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{5} b - 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{2} + 3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{5} b - 7 \, a^{4} b^{2} + 4 \, a^{3} b^{3}\right )}} \]

[In]

integrate(1/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 15.33 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.95 \[ \int \frac {1}{a-b+2 a x^2+a x^4} \, dx=\frac {\ln \left (4\,a^3\,b\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}-4\,a^3\,x+\frac {4\,a^4\,b\,x}{a\,b+\sqrt {a\,b^3}}\right )\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}}{4}+\frac {\ln \left (4\,a^3\,x-4\,a^3\,b\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}-\frac {4\,a^4\,b\,x}{a\,b-\sqrt {a\,b^3}}\right )\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}}{4}-\ln \left (4\,a^3\,x+4\,a^3\,b\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}-\frac {4\,a^4\,b\,x}{a\,b+\sqrt {a\,b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {a\,b^3}}{16\,\left (a\,b^3-a^2\,b^2\right )}}-\ln \left (4\,a^3\,x+16\,a^3\,b\,\sqrt {-\frac {1}{16\,a\,b-16\,\sqrt {a\,b^3}}}-\frac {4\,a^4\,b\,x}{a\,b-\sqrt {a\,b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {a\,b^3}}{16\,\left (a\,b^3-a^2\,b^2\right )}} \]

[In]

int(1/(a - b + 2*a*x^2 + a*x^4),x)

[Out]

(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)