3.157 \(\int \frac{1}{x^7 (3+4 x^3+x^6)} \, dx\)

Optimal. Leaf size=41 \[ \frac{4}{27 x^3}-\frac{1}{18 x^6}-\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{162} \log \left (x^3+3\right )+\frac{13 \log (x)}{27} \]

[Out]

-1/(18*x^6) + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162

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Rubi [A]  time = 0.03602, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1357, 709, 800} \[ \frac{4}{27 x^3}-\frac{1}{18 x^6}-\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{162} \log \left (x^3+3\right )+\frac{13 \log (x)}{27} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(18*x^6) + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 709

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (3+4 x^3+x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^3 \left (3+4 x+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac{1}{18 x^6}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{-4-x}{x^2 \left (3+4 x+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac{1}{18 x^6}+\frac{1}{9} \operatorname{Subst}\left (\int \left (-\frac{4}{3 x^2}+\frac{13}{9 x}-\frac{3}{2 (1+x)}+\frac{1}{18 (3+x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{18 x^6}+\frac{4}{27 x^3}+\frac{13 \log (x)}{27}-\frac{1}{6} \log \left (1+x^3\right )+\frac{1}{162} \log \left (3+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0057354, size = 41, normalized size = 1. \[ \frac{4}{27 x^3}-\frac{1}{18 x^6}-\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{162} \log \left (x^3+3\right )+\frac{13 \log (x)}{27} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(18*x^6) + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162

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Maple [A]  time = 0.009, size = 41, normalized size = 1. \begin{align*} -{\frac{1}{18\,{x}^{6}}}+{\frac{4}{27\,{x}^{3}}}+{\frac{13\,\ln \left ( x \right ) }{27}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+3 \right ) }{162}}-{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^6+4*x^3+3),x)

[Out]

-1/18/x^6+4/27/x^3+13/27*ln(x)-1/6*ln(x^2-x+1)+1/162*ln(x^3+3)-1/6*ln(1+x)

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Maxima [A]  time = 1.05047, size = 47, normalized size = 1.15 \begin{align*} \frac{8 \, x^{3} - 3}{54 \, x^{6}} + \frac{1}{162} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) + \frac{13}{81} \, \log \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/54*(8*x^3 - 3)/x^6 + 1/162*log(x^3 + 3) - 1/6*log(x^3 + 1) + 13/81*log(x^3)

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Fricas [A]  time = 1.48597, size = 109, normalized size = 2.66 \begin{align*} \frac{x^{6} \log \left (x^{3} + 3\right ) - 27 \, x^{6} \log \left (x^{3} + 1\right ) + 78 \, x^{6} \log \left (x\right ) + 24 \, x^{3} - 9}{162 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/162*(x^6*log(x^3 + 3) - 27*x^6*log(x^3 + 1) + 78*x^6*log(x) + 24*x^3 - 9)/x^6

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Sympy [A]  time = 0.20182, size = 34, normalized size = 0.83 \begin{align*} \frac{13 \log{\left (x \right )}}{27} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{162} + \frac{8 x^{3} - 3}{54 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**6+4*x**3+3),x)

[Out]

13*log(x)/27 - log(x**3 + 1)/6 + log(x**3 + 3)/162 + (8*x**3 - 3)/(54*x**6)

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Giac [A]  time = 1.0856, size = 55, normalized size = 1.34 \begin{align*} -\frac{13 \, x^{6} - 8 \, x^{3} + 3}{54 \, x^{6}} + \frac{1}{162} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) + \frac{13}{27} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

-1/54*(13*x^6 - 8*x^3 + 3)/x^6 + 1/162*log(abs(x^3 + 3)) - 1/6*log(abs(x^3 + 1)) + 13/27*log(abs(x))