3.14.56 \(\int \frac {1}{x^4 (1-x^6)} \, dx\) [1356]

Optimal. Leaf size=16 \[ -\frac {1}{3 x^3}+\frac {1}{3} \tanh ^{-1}\left (x^3\right ) \]

[Out]

-1/3/x^3+1/3*arctanh(x^3)

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 331, 212} \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (x^3\right )-\frac {1}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(1 - x^6)),x]

[Out]

-1/3*1/x^3 + ArcTanh[x^3]/3

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 281

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 331

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.88 \begin {gather*} -\frac {1}{3 x^3}-\frac {1}{6} \log \left (1-x^3\right )+\frac {1}{6} \log \left (1+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(1 - x^6)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
time = 0.17, size = 39, normalized size = 2.44

method	result	size
meijerg	\(\frac {i \left (\frac {2 i}{x^{3}}-2 i \arctanh \left (x^{3}\right )\right )}{6}\)	\(18\)
risch	\(-\frac {1}{3 x^{3}}-\frac {\ln \left (x^{3}-1\right )}{6}+\frac {\ln \left (x^{3}+1\right )}{6}\)	\(23\)
default	\(\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\ln \left (x -1\right )}{6}-\frac {1}{3 x^{3}}+\frac {\ln \left (x^{2}-x +1\right )}{6}\)	\(39\)
norman	\(\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}-\frac {\ln \left (x -1\right )}{6}-\frac {1}{3 x^{3}}+\frac {\ln \left (x^{2}-x +1\right )}{6}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 22, normalized size = 1.38 \begin {gather*} -\frac {1}{3 \, x^{3}} + \frac {1}{6} \, \log \left (x^{3} + 1\right ) - \frac {1}{6} \, \log \left (x^{3} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^6+1),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).
time = 0.37, size = 28, normalized size = 1.75 \begin {gather*} \frac {x^{3} \log \left (x^{3} + 1\right ) - x^{3} \log \left (x^{3} - 1\right ) - 2}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^6+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.05, size = 22, normalized size = 1.38 \begin {gather*} - \frac {\log {\left (x^{3} - 1 \right )}}{6} + \frac {\log {\left (x^{3} + 1 \right )}}{6} - \frac {1}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(-x**6+1),x)

[Out]

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

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Giac [A]
time = 1.34, size = 24, normalized size = 1.50 \begin {gather*} -\frac {1}{3 \, x^{3}} + \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^6+1),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (x^3\right )}{3}-\frac {1}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^4*(x^6 - 1)),x)

[Out]

atanh(x^3)/3 - 1/(3*x^3)

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