3.15.74 \(\int \frac {x^3}{1-x^8} \, dx\) [1474]

Optimal. Leaf size=8 \[ \frac {1}{4} \tanh ^{-1}\left (x^4\right ) \]

[Out]

1/4*arctanh(x^4)

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 212} \begin {gather*} \frac {1}{4} \tanh ^{-1}\left (x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(1 - x^8),x]

[Out]

ArcTanh[x^4]/4

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]

Rubi steps

\begin {align*} \int \frac {x^3}{1-x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \tanh ^{-1}\left (x^4\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).
time = 0.00, size = 23, normalized size = 2.88 \begin {gather*} -\frac {1}{8} \log \left (1-x^4\right )+\frac {1}{8} \log \left (1+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 - x^8),x]

[Out]

-1/8*Log[1 - x^4] + Log[1 + x^4]/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(6)=12\).
time = 0.18, size = 30, normalized size = 3.75

method	result	size
meijerg	\(\frac {\arctanh \left (x^{4}\right )}{4}\)	\(7\)
risch	\(-\frac {\ln \left (x^{4}-1\right )}{8}+\frac {\ln \left (x^{4}+1\right )}{8}\)	\(18\)
default	\(-\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{2}+1\right )}{8}+\frac {\ln \left (x^{4}+1\right )}{8}\)	\(30\)
norman	\(-\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{2}+1\right )}{8}+\frac {\ln \left (x^{4}+1\right )}{8}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-x^8+1),x,method=_RETURNVERBOSE)

[Out]

-1/8*ln(x-1)-1/8*ln(x+1)-1/8*ln(x^2+1)+1/8*ln(x^4+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.28, size = 17, normalized size = 2.12 \begin {gather*} \frac {1}{8} \, \log \left (x^{4} + 1\right ) - \frac {1}{8} \, \log \left (x^{4} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^8+1),x, algorithm="maxima")

[Out]

1/8*log(x^4 + 1) - 1/8*log(x^4 - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.35, size = 17, normalized size = 2.12 \begin {gather*} \frac {1}{8} \, \log \left (x^{4} + 1\right ) - \frac {1}{8} \, \log \left (x^{4} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^8+1),x, algorithm="fricas")

[Out]

1/8*log(x^4 + 1) - 1/8*log(x^4 - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
time = 0.04, size = 15, normalized size = 1.88 \begin {gather*} - \frac {\log {\left (x^{4} - 1 \right )}}{8} + \frac {\log {\left (x^{4} + 1 \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(-x**8+1),x)

[Out]

-log(x**4 - 1)/8 + log(x**4 + 1)/8

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (6) = 12\).
time = 1.69, size = 18, normalized size = 2.25 \begin {gather*} \frac {1}{8} \, \log \left (x^{4} + 1\right ) - \frac {1}{8} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^8+1),x, algorithm="giac")

[Out]

1/8*log(x^4 + 1) - 1/8*log(abs(x^4 - 1))

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Mupad [B]
time = 0.07, size = 6, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (x^4\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^3/(x^8 - 1),x)

[Out]

atanh(x^4)/4

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