3.8.7 \(\int \frac {x^2}{3+x^4} \, dx\) [707]

Optimal. Leaf size=133 \[ -\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\log \left (\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2\right )}{4 \sqrt {2} \sqrt [4]{3}} \]

[Out]

1/12*arctan(-1+1/3*x*2^(1/2)*3^(3/4))*3^(3/4)*2^(1/2)+1/12*arctan(1+1/3*x*2^(1/2)*3^(3/4))*3^(3/4)*2^(1/2)+1/2
4*ln(x^2-3^(1/4)*x*2^(1/2)+3^(1/2))*3^(3/4)*2^(1/2)-1/24*ln(x^2+3^(1/4)*x*2^(1/2)+3^(1/2))*3^(3/4)*2^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\log \left (x^2-\sqrt {2} \sqrt [4]{3} x+\sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\log \left (x^2+\sqrt {2} \sqrt [4]{3} x+\sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(3 + x^4),x]

[Out]

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

Rule 210

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

Rule 303

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

Rule 631

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 642

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]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 101, normalized size = 0.76 \begin {gather*} \frac {-2 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )+2 \tan ^{-1}\left (1+\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )+\log \left (3-\sqrt {2} 3^{3/4} x+\sqrt {3} x^2\right )-\log \left (3+\sqrt {2} 3^{3/4} x+\sqrt {3} x^2\right )}{4 \sqrt {2} \sqrt [4]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(3 + x^4),x]

[Out]

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

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Maple [A]
time = 0.14, size = 73, normalized size = 0.55

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+3\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{4}\)	\(22\)
default	\(\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-3^{\frac {1}{4}} x \sqrt {2}+\sqrt {3}}{x^{2}+3^{\frac {1}{4}} x \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {x \sqrt {2}\, 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {x \sqrt {2}\, 3^{\frac {3}{4}}}{3}\right )\right )}{24}\)	\(73\)
meijerg	\(\frac {3^{\frac {3}{4}} \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{3}+\frac {\sqrt {3}\, \sqrt {x^{4}}}{3}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{6-\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{3}+\frac {\sqrt {3}\, \sqrt {x^{4}}}{3}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{6+\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{12}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^4+3),x,method=_RETURNVERBOSE)

[Out]

1/24*3^(3/4)*2^(1/2)*(ln((x^2-3^(1/4)*x*2^(1/2)+3^(1/2))/(x^2+3^(1/4)*x*2^(1/2)+3^(1/2)))+2*arctan(1+1/3*x*2^(
1/2)*3^(3/4))+2*arctan(-1+1/3*x*2^(1/2)*3^(3/4)))

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Maxima [A]
time = 0.51, size = 107, normalized size = 0.80 \begin {gather*} \frac {1}{12} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (x^{2} + 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (x^{2} - 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+3),x, algorithm="maxima")

[Out]

1/12*3^(3/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x + 3^(1/4)*sqrt(2))) + 1/12*3^(3/4)*sqrt(2)*arctan(1/6*3^(
3/4)*sqrt(2)*(2*x - 3^(1/4)*sqrt(2))) - 1/24*3^(3/4)*sqrt(2)*log(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*3^(
3/4)*sqrt(2)*log(x^2 - 3^(1/4)*sqrt(2)*x + sqrt(3))

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Fricas [A]
time = 0.39, size = 150, normalized size = 1.13 \begin {gather*} -\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} x + \frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {x^{2} + 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}} - 1\right ) - \frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} x + \frac {1}{3} \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {x^{2} - 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}} + 1\right ) - \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (4 \, x^{2} + 4 \cdot 3^{\frac {1}{4}} \sqrt {2} x + 4 \, \sqrt {3}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (4 \, x^{2} - 4 \cdot 3^{\frac {1}{4}} \sqrt {2} x + 4 \, \sqrt {3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+3),x, algorithm="fricas")

[Out]

-1/6*3^(3/4)*sqrt(2)*arctan(-1/3*3^(3/4)*sqrt(2)*x + 1/3*3^(3/4)*sqrt(2)*sqrt(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3
)) - 1) - 1/6*3^(3/4)*sqrt(2)*arctan(-1/3*3^(3/4)*sqrt(2)*x + 1/3*3^(3/4)*sqrt(2)*sqrt(x^2 - 3^(1/4)*sqrt(2)*x
+ sqrt(3)) + 1) - 1/24*3^(3/4)*sqrt(2)*log(4*x^2 + 4*3^(1/4)*sqrt(2)*x + 4*sqrt(3)) + 1/24*3^(3/4)*sqrt(2)*lo
g(4*x^2 - 4*3^(1/4)*sqrt(2)*x + 4*sqrt(3))

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Sympy [A]
time = 0.18, size = 124, normalized size = 0.93 \begin {gather*} \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (x^{2} - \sqrt {2} \cdot \sqrt [4]{3} x + \sqrt {3} \right )}}{24} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (x^{2} + \sqrt {2} \cdot \sqrt [4]{3} x + \sqrt {3} \right )}}{24} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x}{3} - 1 \right )}}{12} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x}{3} + 1 \right )}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**4+3),x)

[Out]

sqrt(2)*3**(3/4)*log(x**2 - sqrt(2)*3**(1/4)*x + sqrt(3))/24 - sqrt(2)*3**(3/4)*log(x**2 + sqrt(2)*3**(1/4)*x
+ sqrt(3))/24 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*x/3 - 1)/12 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*x/
3 + 1)/12

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Giac [A]
time = 0.49, size = 95, normalized size = 0.71 \begin {gather*} \frac {1}{12} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{12} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {1}{24} \cdot 108^{\frac {1}{4}} \log \left (x^{2} + 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) + \frac {1}{24} \cdot 108^{\frac {1}{4}} \log \left (x^{2} - 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+3),x, algorithm="giac")

[Out]

1/12*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x + 3^(1/4)*sqrt(2))) + 1/12*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)
*(2*x - 3^(1/4)*sqrt(2))) - 1/24*108^(1/4)*log(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*108^(1/4)*log(x^2 - 3
^(1/4)*sqrt(2)*x + sqrt(3))

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Mupad [B]
time = 0.11, size = 45, normalized size = 0.34 \begin {gather*} \sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,x\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,x\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^4 + 3),x)

[Out]

2^(1/2)*3^(3/4)*atan(2^(1/2)*3^(3/4)*x*(1/6 - 1i/6))*(1/12 - 1i/12) + 2^(1/2)*3^(3/4)*atan(2^(1/2)*3^(3/4)*x*(
1/6 + 1i/6))*(1/12 + 1i/12)

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