3.2.24 \(\int \frac {1}{x^4 (a^3+x^3)} \, dx\) [124]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 = {272, 46} \begin {gather*} -\frac {\log (x)}{a^6}-\frac {1}{3 a^3 x^3}+\frac {\log \left (a^3+x^3\right )}{3 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a^3 + x^3)),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 272

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]]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} -\frac {1}{3 a^3 x^3}-\frac {\log (x)}{a^6}+\frac {\log \left (a^3+x^3\right )}{3 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a^3 + x^3)),x]

[Out]

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

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Maple [A]
time = 0.05, size = 43, normalized size = 1.30

method	result	size
risch	\(-\frac {1}{3 a^{3} x^{3}}-\frac {\ln \left (x \right )}{a^{6}}+\frac {\ln \left (-a^{3}-x^{3}\right )}{3 a^{6}}\)	\(34\)
default	\(-\frac {1}{3 a^{3} x^{3}}-\frac {\ln \left (x \right )}{a^{6}}+\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{6}}+\frac {\ln \left (a +x \right )}{3 a^{6}}\)	\(43\)
norman	\(-\frac {1}{3 a^{3} x^{3}}-\frac {\ln \left (x \right )}{a^{6}}+\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{6}}+\frac {\ln \left (a +x \right )}{3 a^{6}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 2.46, size = 31, normalized size = 0.94 \begin {gather*} \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 {gather*}

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 = 0.38, size = 33, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.11, size = 29, normalized size = 0.88 \begin {gather*} - \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 {gather*}

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 = 1.00, size = 40, normalized size = 1.21 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} \frac {\ln \left (a^3+x^3\right )}{3\,a^6}-\frac {\ln \left (x\right )}{a^6}-\frac {1}{3\,a^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a^3 + x^3)),x)

[Out]

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

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