3.1362 \(\int \frac{x^7}{1+x^6} \, dx\)

Optimal. Leaf size=56 \[ \frac{x^2}{2}-\frac{1}{6} \log \left (x^2+1\right )+\frac{1}{12} \log \left (x^4-x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0428234, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {275, 321, 200, 31, 634, 618, 204, 628} \[ \frac{x^2}{2}-\frac{1}{6} \log \left (x^2+1\right )+\frac{1}{12} \log \left (x^4-x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 + x^6),x]

[Out]

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
 + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
 c, n, m, p, x]

Rule 200

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 31

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

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

Mathematica [A]  time = 0.0141678, size = 79, normalized size = 1.41 \[ \frac{1}{12} \left (6 x^2-2 \log \left (x^2+1\right )+\log \left (x^2-\sqrt{3} x+1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+2 \sqrt{3} \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 + x^6),x]

[Out]

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

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^6+1),x)

[Out]

1/2*x^2+1/12*ln(x^4-x^2+1)-1/6*3^(1/2)*arctan(1/3*(2*x^2-1)*3^(1/2))-1/6*ln(x^2+1)

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Maxima [A]  time = 1.50142, size = 61, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46882, size = 135, normalized size = 2.41 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.132635, size = 51, normalized size = 0.91 \begin{align*} \frac{x^{2}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} - \frac{\sqrt{3}}{3} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**6+1),x)

[Out]

x**2/2 - log(x**2 + 1)/6 + log(x**4 - x**2 + 1)/12 - sqrt(3)*atan(2*sqrt(3)*x**2/3 - sqrt(3)/3)/6

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Giac [A]  time = 1.12547, size = 61, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^6+1),x, algorithm="giac")

[Out]

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