3.160 \(\int \frac {x^7}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=119 \[ \frac {x^2}{2}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {1}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\frac {1}{6} \log (x+1)+\frac {1}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {3}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

[Out]

1/2*x^2+3/2*3^(1/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/2*3^(2/3)*ln(3^(1/3)+x)+1/12*ln(x^2-x+1)-1
/4*3^(2/3)*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 = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1367, 1510, 292, 31, 634, 618, 204, 628, 617} \[ \frac {x^2}{2}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {1}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\frac {1}{6} \log (x+1)+\frac {1}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {3}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(3 + 4*x^3 + x^6),x]

[Out]

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

Rule 31

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

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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 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 618

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 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]

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 1367

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(d^(2*n - 1)*(d*x)
^(m - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(m + 2*n*p + 1)), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), In
t[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; Fr
eeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n
*p + 1, 0] && IntegerQ[p]

Rule 1510

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

Rubi steps

\begin {align*} \int \frac {x^7}{3+4 x^3+x^6} \, dx &=\frac {x^2}{2}-\frac {1}{2} \int \frac {x \left (6+8 x^3\right )}{3+4 x^3+x^6} \, dx\\ &=\frac {x^2}{2}+\frac {1}{2} \int \frac {x}{1+x^3} \, dx-\frac {9}{2} \int \frac {x}{3+x^3} \, dx\\ &=\frac {x^2}{2}-\frac {1}{6} \int \frac {1}{1+x} \, dx+\frac {1}{6} \int \frac {1+x}{1-x+x^2} \, dx+\frac {1}{2} 3^{2/3} \int \frac {1}{\sqrt [3]{3}+x} \, dx-\frac {1}{2} 3^{2/3} \int \frac {\sqrt [3]{3}+x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=\frac {x^2}{2}-\frac {1}{6} \log (1+x)+\frac {1}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )+\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 {9}{4} \int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx-\frac {1}{4} 3^{2/3} \int \frac {-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx\\ &=\frac {x^2}{2}-\frac {1}{6} \log (1+x)+\frac {1}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {1}{4} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {1}{2} \left (3\ 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{3}}\right )\\ &=\frac {x^2}{2}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {3}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-\frac {1}{6} \log (1+x)+\frac {1}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {1}{4} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 111, normalized size = 0.93 \[ \frac {1}{12} \left (6 x^2+\log \left (x^2-x+1\right )-3\ 3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-2 \log (x+1)+6\ 3^{2/3} \log \left (3^{2/3} x+3\right )+18 \sqrt [6]{3} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(3 + 4*x^3 + x^6),x]

[Out]

(6*x^2 + 18*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + 6*
3^(2/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3*3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12

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fricas [A]  time = 1.12, size = 99, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \cdot 9^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {2}{9} \cdot 9^{\frac {1}{3}} \sqrt {3} x - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{4} \cdot 9^{\frac {1}{3}} \log \left (3 \, x^{2} - 9^{\frac {2}{3}} x + 3 \cdot 9^{\frac {1}{3}}\right ) + \frac {1}{2} \cdot 9^{\frac {1}{3}} \log \left (3 \, x + 9^{\frac {2}{3}}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/2*x^2 - 1/2*9^(1/3)*sqrt(3)*arctan(2/9*9^(1/3)*sqrt(3)*x - 1/3*sqrt(3)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*
x - 1)) - 1/4*9^(1/3)*log(3*x^2 - 9^(2/3)*x + 3*9^(1/3)) + 1/2*9^(1/3)*log(3*x + 9^(2/3)) + 1/12*log(x^2 - x +
1) - 1/6*log(x + 1)

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giac [A]  time = 0.34, size = 91, normalized size = 0.76 \[ \frac {1}{2} \, x^{2} - \frac {1}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 89, normalized size = 0.75 \[ \frac {x^{2}}{2}-\frac {3 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {3^{\frac {2}{3}} \ln \left (x +3^{\frac {1}{3}}\right )}{2}-\frac {3^{\frac {2}{3}} \ln \left (x^{2}-3^{\frac {1}{3}} x +3^{\frac {2}{3}}\right )}{4}+\frac {\ln \left (x^{2}-x +1\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^6+4*x^3+3),x)

[Out]

1/2*x^2-1/6*ln(x+1)+1/2*3^(2/3)*ln(x+3^(1/3))-1/4*3^(2/3)*ln(x^2-3^(1/3)*x+3^(2/3))-3/2*3^(1/6)*arctan(1/3*3^(
1/2)*(2/3*3^(2/3)*x-1))+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 = 1.31, size = 89, normalized size = 0.75 \[ \frac {1}{2} \, x^{2} - \frac {1}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 118, normalized size = 0.99 \[ \frac {3^{2/3}\,\ln \left (x+3^{1/3}\right )}{2}-\frac {\ln \left (x+1\right )}{6}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {x^2}{2}-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{4}-\frac {3^{1/6}\,3{}\mathrm {i}}{4}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{4}+\frac {3^{1/6}\,3{}\mathrm {i}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(4*x^3 + x^6 + 3),x)

[Out]

(3^(2/3)*log(x + 3^(1/3)))/2 - 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) + x^2/2 - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*(3^(2/3)/4 - (3^
(1/6)*3i)/4) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3^(2/3)/4 + (3^(1/6)*3i)/4)

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sympy [C]  time = 0.61, size = 134, normalized size = 1.13 \[ \frac {x^{2}}{2} - \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {6562 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{183} - \frac {1872 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{61} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {1872 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{61} + \frac {6562 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{183} \right )} + \operatorname {RootSum} {\left (8 t^{3} - 9, \left (t \mapsto t \log {\left (- \frac {1872 t^{5}}{61} + \frac {6562 t^{2}}{183} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**6+4*x**3+3),x)

[Out]

x**2/2 - log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 6562*(1/12 - sqrt(3)*I/12)**2/183 - 1872*(1/12 - sqrt(3)
*I/12)**5/61) + (1/12 + sqrt(3)*I/12)*log(x - 1872*(1/12 + sqrt(3)*I/12)**5/61 + 6562*(1/12 + sqrt(3)*I/12)**2
/183) + RootSum(8*_t**3 - 9, Lambda(_t, _t*log(-1872*_t**5/61 + 6562*_t**2/183 + x)))

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