3.124 \(\int \frac{1}{x^4 (a^3+x^3)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{3 a^3 x^3}+\frac{\log \left (a^3+x^3\right )}{3 a^6}-\frac{\log (x)}{a^6} \]

[Out]

-1/(3*a^3*x^3) - Log[x]/a^6 + Log[a^3 + x^3]/(3*a^6)

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Rubi [A]  time = 0.0195942, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac{1}{3 a^3 x^3}+\frac{\log \left (a^3+x^3\right )}{3 a^6}-\frac{\log (x)}{a^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a^3 + x^3)),x]

[Out]

-1/(3*a^3*x^3) - Log[x]/a^6 + Log[a^3 + x^3]/(3*a^6)

Rule 266

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0052609, size = 33, normalized size = 1. \[ -\frac{1}{3 a^3 x^3}+\frac{\log \left (a^3+x^3\right )}{3 a^6}-\frac{\log (x)}{a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a^3 + x^3)),x]

[Out]

-1/(3*a^3*x^3) - Log[x]/a^6 + Log[a^3 + x^3]/(3*a^6)

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Maple [A]  time = 0.006, size = 43, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,{a}^{3}{x}^{3}}}-{\frac{\ln \left ( x \right ) }{{a}^{6}}}+{\frac{\ln \left ({a}^{2}-ax+{x}^{2} \right ) }{3\,{a}^{6}}}+{\frac{\ln \left ( a+x \right ) }{3\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a^3+x^3),x)

[Out]

-1/3/a^3/x^3-ln(x)/a^6+1/3/a^6*ln(a^2-a*x+x^2)+1/3*ln(a+x)/a^6

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Maxima [A]  time = 0.925473, size = 42, normalized size = 1.27 \begin{align*} \frac{\log \left (a^{3} + x^{3}\right )}{3 \, a^{6}} - \frac{\log \left (x^{3}\right )}{3 \, a^{6}} - \frac{1}{3 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^3+x^3),x, algorithm="maxima")

[Out]

1/3*log(a^3 + x^3)/a^6 - 1/3*log(x^3)/a^6 - 1/3/(a^3*x^3)

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Fricas [A]  time = 2.037, size = 77, normalized size = 2.33 \begin{align*} \frac{x^{3} \log \left (a^{3} + x^{3}\right ) - 3 \, x^{3} \log \left (x\right ) - a^{3}}{3 \, a^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^3+x^3),x, algorithm="fricas")

[Out]

1/3*(x^3*log(a^3 + x^3) - 3*x^3*log(x) - a^3)/(a^6*x^3)

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Sympy [A]  time = 0.391197, size = 29, normalized size = 0.88 \begin{align*} - \frac{1}{3 a^{3} x^{3}} - \frac{\log{\left (x \right )}}{a^{6}} + \frac{\log{\left (a^{3} + x^{3} \right )}}{3 a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a**3+x**3),x)

[Out]

-1/(3*a**3*x**3) - log(x)/a**6 + log(a**3 + x**3)/(3*a**6)

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Giac [A]  time = 1.05288, size = 54, normalized size = 1.64 \begin{align*} \frac{\log \left ({\left | a^{3} + x^{3} \right |}\right )}{3 \, a^{6}} - \frac{\log \left ({\left | x \right |}\right )}{a^{6}} - \frac{a^{3} - x^{3}}{3 \, a^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a^3+x^3),x, algorithm="giac")

[Out]

1/3*log(abs(a^3 + x^3))/a^6 - log(abs(x))/a^6 - 1/3*(a^3 - x^3)/(a^6*x^3)