3.714 \(\int \frac{x^2}{2 (a+b)+x^4} \, dx\)

Optimal. Leaf size=79 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]

[Out]

ArcTan[x/(2^(1/4)*(-a - b)^(1/4))]/(2*2^(1/4)*(-a - b)^(1/4)) - ArcTanh[x/(2^(1/4)*(-a - b)^(1/4))]/(2*2^(1/4)
*(-a - b)^(1/4))

________________________________________________________________________________________

Rubi [A]  time = 0.0242379, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {298, 203, 206} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(2*(a + b) + x^4),x]

[Out]

ArcTan[x/(2^(1/4)*(-a - b)^(1/4))]/(2*2^(1/4)*(-a - b)^(1/4)) - ArcTanh[x/(2^(1/4)*(-a - b)^(1/4))]/(2*2^(1/4)
*(-a - b)^(1/4))

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0112943, size = 128, normalized size = 1.62 \[ \frac{\log \left (-2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-\log \left (2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}+1\right )}{4\ 2^{3/4} \sqrt [4]{a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(2*(a + b) + x^4),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0., size = 137, normalized size = 1.7 \begin{align*}{\frac{\sqrt{2}}{8}\ln \left ({ \left ({x}^{2}-\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) \left ({x}^{2}+\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}-1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/(2*a+2*b)^(1/4)*2^(1/2)*ln((x^2-(2*a+2*b)^(1/4)*x*2^(1/2)+(2*a+2*b)^(1/2))/(x^2+(2*a+2*b)^(1/4)*x*2^(1/2)+
(2*a+2*b)^(1/2)))+1/4/(2*a+2*b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(2*a+2*b)^(1/4)*x+1)+1/4/(2*a+2*b)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(2*a+2*b)^(1/4)*x-1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.52447, size = 448, normalized size = 5.67 \begin{align*} -\left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \arctan \left (-\left (\frac{1}{2}\right )^{\frac{1}{4}} x \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} + \left (\frac{1}{2}\right )^{\frac{1}{4}} \sqrt{x^{2} - 2 \, \sqrt{\frac{1}{2}}{\left (a + b\right )} \sqrt{-\frac{1}{a + b}}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}}\right ) + \frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \log \left (2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}} + x\right ) - \frac{1}{4} \, \left (\frac{1}{2}\right )^{\frac{1}{4}} \left (-\frac{1}{a + b}\right )^{\frac{1}{4}} \log \left (-2 \, \left (\frac{1}{2}\right )^{\frac{3}{4}}{\left (a + b\right )} \left (-\frac{1}{a + b}\right )^{\frac{3}{4}} + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/2)^(1/4)*(-1/(a + b))^(1/4)*arctan(-(1/2)^(1/4)*x*(-1/(a + b))^(1/4) + (1/2)^(1/4)*sqrt(x^2 - 2*sqrt(1/2)*
(a + b)*sqrt(-1/(a + b)))*(-1/(a + b))^(1/4)) + 1/4*(1/2)^(1/4)*(-1/(a + b))^(1/4)*log(2*(1/2)^(3/4)*(a + b)*(
-1/(a + b))^(3/4) + x) - 1/4*(1/2)^(1/4)*(-1/(a + b))^(1/4)*log(-2*(1/2)^(3/4)*(a + b)*(-1/(a + b))^(3/4) + x)

________________________________________________________________________________________

Sympy [A]  time = 0.212837, size = 29, normalized size = 0.37 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (512 a + 512 b\right ) + 1, \left ( t \mapsto t \log{\left (128 t^{3} a + 128 t^{3} b + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**4*(512*a + 512*b) + 1, Lambda(_t, _t*log(128*_t**3*a + 128*_t**3*b + x)))

________________________________________________________________________________________

Giac [B]  time = 1.11621, size = 296, normalized size = 3.75 \begin{align*} \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}}}\right )}{4 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} - \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} + \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2}{\left (2 \, a + 2 \, b\right )}^{\frac{1}{4}} x + \sqrt{2 \, a + 2 \, b}\right )}{8 \,{\left (\sqrt{2} a + \sqrt{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a + 2*b)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(2*a + 2*b)^(1/4))/(2*a + 2*b)^(1/4))/(sqrt(2)*a + sqr
t(2)*b) + 1/4*(2*a + 2*b)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(2*a + 2*b)^(1/4))/(2*a + 2*b)^(1/4))/(sqrt(
2)*a + sqrt(2)*b) - 1/8*(2*a + 2*b)^(3/4)*log(x^2 + sqrt(2)*(2*a + 2*b)^(1/4)*x + sqrt(2*a + 2*b))/(sqrt(2)*a
+ sqrt(2)*b) + 1/8*(2*a + 2*b)^(3/4)*log(x^2 - sqrt(2)*(2*a + 2*b)^(1/4)*x + sqrt(2*a + 2*b))/(sqrt(2)*a + sqr
t(2)*b)