3.2.63 \(\int \frac {x^3}{3+4 x^3+x^6} \, dx\) [163]

Optimal. Leaf size=112 \[ \frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4\ 3^{2/3}} \]

[Out]

-1/6*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/6*3^(1/3)*ln(3^(1/3)+x)+1/12*ln(x^2-x+1)-1/12*3^(
1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)+1/6*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1388, 206, 31, 648, 631, 210, 642, 632} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4\ 3^{2/3}}-\frac {1}{6} \log (x+1)+\frac {\log \left (x+\sqrt [3]{3}\right )}{2\ 3^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(3 + 4*x^3 + x^6),x]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(3^(1/3) - 2*x)/3^(5/6)]/(2*3^(1/6)) - Log[1 + x]/6 + Log[3^(1/
3) + x]/(2*3^(2/3)) + Log[1 - x + x^2]/12 - Log[3^(2/3) - 3^(1/3)*x + x^2]/(4*3^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

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 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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 648

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 1388

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n/2)*(b/q + 1), Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n/2)*(b/q - 1), Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rubi steps

\begin {align*} \int \frac {x^3}{3+4 x^3+x^6} \, dx &=-\left (\frac {1}{2} \int \frac {1}{1+x^3} \, dx\right )+\frac {3}{2} \int \frac {1}{3+x^3} \, dx\\ &=-\left (\frac {1}{6} \int \frac {1}{1+x} \, dx\right )-\frac {1}{6} \int \frac {2-x}{1-x+x^2} \, dx+\frac {\int \frac {1}{\sqrt [3]{3}+x} \, dx}{2\ 3^{2/3}}+\frac {\int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{2\ 3^{2/3}}\\ &=-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac {1}{12} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {1}{4} \int \frac {1}{1-x+x^2} \, dx-\frac {\int \frac {-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{4\ 3^{2/3}}+\frac {1}{4} 3^{2/3} \int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4\ 3^{2/3}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} \sqrt [3]{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{3}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4\ 3^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 106, normalized size = 0.95 \begin {gather*} \frac {1}{12} \left (-2 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \log (1+x)+2 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )+\log \left (1-x+x^2\right )-\sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(3 + 4*x^3 + x^6),x]

[Out]

(-2*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + 2*3^(1/3)*
Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12

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Maple [A]
time = 0.03, size = 84, normalized size = 0.75

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (9 \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (x +3 \textit {\_R} \right )\right )}{2}-\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\)	\(54\)
default	\(\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{12}+\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{6}-\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^6+4*x^3+3),x,method=_RETURNVERBOSE)

[Out]

1/6*3^(1/3)*ln(3^(1/3)+x)-1/12*3^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)+1/6*3^(5/6)*arctan(1/3*3^(1/2)*(2/3*3^(2/3)*x
-1))-1/6*ln(1+x)+1/12*ln(x^2-x+1)-1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]
time = 0.50, size = 84, normalized size = 0.75 \begin {gather*} \frac {1}{6} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/6*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/12*3^(1/3)*log
(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/6*3^(1/3)*log(x + 3^(1/3)) + 1/12*log(x^2 - x + 1) - 1/6*log(x + 1)

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Fricas [A]
time = 0.36, size = 102, normalized size = 0.91 \begin {gather*} \frac {1}{6} \cdot 9^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{36} \cdot 9^{\frac {2}{3}} \log \left (3 \, x^{2} - 9^{\frac {2}{3}} x + 3 \cdot 9^{\frac {1}{3}}\right ) + \frac {1}{18} \cdot 9^{\frac {2}{3}} \log \left (3 \, x + 9^{\frac {2}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/6*9^(1/6)*sqrt(3)*arctan(1/27*9^(1/6)*(2*9^(2/3)*sqrt(3)*x - 3*9^(1/3)*sqrt(3))) - 1/36*9^(2/3)*log(3*x^2 -
9^(2/3)*x + 3*9^(1/3)) + 1/18*9^(2/3)*log(3*x + 9^(2/3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/12*lo
g(x^2 - x + 1) - 1/6*log(x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.30, size = 110, normalized size = 0.98 \begin {gather*} - \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {1}{4} + 648 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4} + \frac {\sqrt {3} i}{4} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {1}{4} + 648 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4} - \frac {\sqrt {3} i}{4} \right )} + \operatorname {RootSum} {\left (72 t^{3} - 1, \left ( t \mapsto t \log {\left (648 t^{4} - 3 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**6+4*x**3+3),x)

[Out]

-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x - 1/4 + 648*(1/12 - sqrt(3)*I/12)**4 + sqrt(3)*I/4) + (1/12 + sqrt
(3)*I/12)*log(x - 1/4 + 648*(1/12 + sqrt(3)*I/12)**4 - sqrt(3)*I/4) + RootSum(72*_t**3 - 1, Lambda(_t, _t*log(
648*_t**4 - 3*_t + x)))

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Giac [A]
time = 3.18, size = 86, normalized size = 0.77 \begin {gather*} \frac {1}{6} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

1/6*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/12*3^(1/3)*log
(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/6*3^(1/3)*log(abs(x + 3^(1/3))) + 1/12*log(x^2 - x + 1) - 1/6*log(abs(x + 1))

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Mupad [B]
time = 1.36, size = 113, normalized size = 1.01 \begin {gather*} \frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{6}-\frac {\ln \left (x+1\right )}{6}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{12}+\frac {3^{5/6}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{12}-\frac {3^{5/6}\,1{}\mathrm {i}}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(4*x^3 + x^6 + 3),x)

[Out]

(3^(1/3)*log(x + 3^(1/3)))/6 - log(x + 1)/6 + log(x - (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/12 + 1/12) - log(x +
(3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/12 - 1/12) - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*(3^(1/3)/12 + (3^(5/6)*1
i)/12) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3^(1/3)/12 - (3^(5/6)*1i)/12)

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