3.107 \(\int \frac{2 \sqrt{a}-x^2}{a-\sqrt{a} x^2+x^4} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 \sqrt [4]{a}} \]

[Out]

-ArcTan[Sqrt[3] - (2*x)/a^(1/4)]/(2*a^(1/4)) + ArcTan[Sqrt[3] + (2*x)/a^(1/4)]/(2*a^(1/4)) - (Sqrt[3]*Log[Sqrt
[a] - Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4)) + (Sqrt[3]*Log[Sqrt[a] + Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4))

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Rubi [A]  time = 0.0796739, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {1169, 634, 617, 204, 628} \[ -\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [4]{a} x+\sqrt{a}+x^2\right )}{4 \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [4]{a}}+\sqrt{3}\right )}{2 \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]

Int[(2*Sqrt[a] - x^2)/(a - Sqrt[a]*x^2 + x^4),x]

[Out]

-ArcTan[Sqrt[3] - (2*x)/a^(1/4)]/(2*a^(1/4)) + ArcTan[Sqrt[3] + (2*x)/a^(1/4)]/(2*a^(1/4)) - (Sqrt[3]*Log[Sqrt
[a] - Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4)) + (Sqrt[3]*Log[Sqrt[a] + Sqrt[3]*a^(1/4)*x + x^2])/(4*a^(1/4))

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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]

Rubi steps

\begin{align*} \int \frac{2 \sqrt{a}-x^2}{a-\sqrt{a} x^2+x^4} \, dx &=\frac{\int \frac{2 \sqrt{3} a^{3/4}-3 \sqrt{a} x}{\sqrt{a}-\sqrt{3} \sqrt [4]{a} x+x^2} \, dx}{2 \sqrt{3} a^{3/4}}+\frac{\int \frac{2 \sqrt{3} a^{3/4}+3 \sqrt{a} x}{\sqrt{a}+\sqrt{3} \sqrt [4]{a} x+x^2} \, dx}{2 \sqrt{3} a^{3/4}}\\ &=\frac{1}{4} \int \frac{1}{\sqrt{a}-\sqrt{3} \sqrt [4]{a} x+x^2} \, dx+\frac{1}{4} \int \frac{1}{\sqrt{a}+\sqrt{3} \sqrt [4]{a} x+x^2} \, dx-\frac{\sqrt{3} \int \frac{-\sqrt{3} \sqrt [4]{a}+2 x}{\sqrt{a}-\sqrt{3} \sqrt [4]{a} x+x^2} \, dx}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \int \frac{\sqrt{3} \sqrt [4]{a}+2 x}{\sqrt{a}+\sqrt{3} \sqrt [4]{a} x+x^2} \, dx}{4 \sqrt [4]{a}}\\ &=-\frac{\sqrt{3} \log \left (\sqrt{a}-\sqrt{3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{a}+\sqrt{3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 x}{\sqrt{3} \sqrt [4]{a}}\right )}{2 \sqrt{3} \sqrt [4]{a}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 x}{\sqrt{3} \sqrt [4]{a}}\right )}{2 \sqrt{3} \sqrt [4]{a}}\\ &=-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}+\frac{\tan ^{-1}\left (\sqrt{3}+\frac{2 x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\sqrt{3} \log \left (\sqrt{a}-\sqrt{3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}+\frac{\sqrt{3} \log \left (\sqrt{a}+\sqrt{3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt [4]{a}}\\ \end{align*}

Mathematica [C]  time = 0.157455, size = 115, normalized size = 0.94 \[ \frac{\sqrt [4]{-1} \left (\sqrt{\sqrt{3}-i} \left (\sqrt{3}-3 i\right ) \tanh ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt [4]{a}}\right )-\sqrt{\sqrt{3}+i} \left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt [4]{a}}\right )\right )}{2 \sqrt{6} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*Sqrt[a] - x^2)/(a - Sqrt[a]*x^2 + x^4),x]

[Out]

((-1)^(1/4)*(-(Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*a^(1/4))]) + Sqrt[-I +
Sqrt[3]]*(-3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*a^(1/4))]))/(2*Sqrt[6]*a^(1/4))

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Maple [A]  time = 0.073, size = 96, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{4}\ln \left ({x}^{2}+\sqrt [4]{a}x\sqrt{3}+\sqrt{a} \right ){\frac{1}{\sqrt [4]{a}}}}+{\frac{1}{2}\arctan \left ({ \left ( 2\,x+\sqrt{3}\sqrt [4]{a} \right ){\frac{1}{\sqrt [4]{a}}}} \right ){\frac{1}{\sqrt [4]{a}}}}-{\frac{\sqrt{3}}{4}\ln \left ( \sqrt [4]{a}x\sqrt{3}-{x}^{2}-\sqrt{a} \right ){\frac{1}{\sqrt [4]{a}}}}-{\frac{1}{2}\arctan \left ({ \left ( \sqrt{3}\sqrt [4]{a}-2\,x \right ){\frac{1}{\sqrt [4]{a}}}} \right ){\frac{1}{\sqrt [4]{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(x^2+a^(1/4)*x*3^(1/2)+a^(1/2))*3^(1/2)/a^(1/4)+1/2/a^(1/4)*arctan((2*x+3^(1/2)*a^(1/4))/a^(1/4))-1/4/a^
(1/4)*3^(1/2)*ln(a^(1/4)*x*3^(1/2)-x^2-a^(1/2))-1/2/a^(1/4)*arctan((3^(1/2)*a^(1/4)-2*x)/a^(1/4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 2 \, \sqrt{a}}{x^{4} - \sqrt{a} x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.67147, size = 682, normalized size = 5.59 \begin{align*} \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} \log \left (\sqrt{\frac{1}{2}} \sqrt{a} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} + x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} \log \left (-\sqrt{\frac{1}{2}} \sqrt{a} \sqrt{\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} + \sqrt{a}}{a}} + x\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} \log \left (\sqrt{\frac{1}{2}} \sqrt{a} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} + x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} \log \left (-\sqrt{\frac{1}{2}} \sqrt{a} \sqrt{-\frac{\sqrt{3} a \sqrt{-\frac{1}{a}} - \sqrt{a}}{a}} + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(1/2)*sqrt((sqrt(3)*a*sqrt(-1/a) + sqrt(a))/a)*log(sqrt(1/2)*sqrt(a)*sqrt((sqrt(3)*a*sqrt(-1/a) + sqrt
(a))/a) + x) - 1/2*sqrt(1/2)*sqrt((sqrt(3)*a*sqrt(-1/a) + sqrt(a))/a)*log(-sqrt(1/2)*sqrt(a)*sqrt((sqrt(3)*a*s
qrt(-1/a) + sqrt(a))/a) + x) + 1/2*sqrt(1/2)*sqrt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a)*log(sqrt(1/2)*sqrt(a)*s
qrt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a) + x) - 1/2*sqrt(1/2)*sqrt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a)*log(-s
qrt(1/2)*sqrt(a)*sqrt(-(sqrt(3)*a*sqrt(-1/a) - sqrt(a))/a) + x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: PolynomialError

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError