3.2.31 \(\int \frac {1}{x^3 (a^4-x^4)} \, dx\) [131]

Optimal. Leaf size=26 \[ -\frac {1}{2 a^4 x^2}+\frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^6} \]

[Out]

-1/2/a^4/x^2+1/2*arctanh(x^2/a^2)/a^6

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {281, 331, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^6}-\frac {1}{2 a^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^4 - x^4)),x]

[Out]

-1/2*1/(a^4*x^2) + ArcTanh[x^2/a^2]/(2*a^6)

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

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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.92 \begin {gather*} -\frac {1}{2 a^4 x^2}-\frac {\log (a-x)}{4 a^6}-\frac {\log (a+x)}{4 a^6}+\frac {\log \left (a^2+x^2\right )}{4 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a^4 - x^4)),x]

[Out]

-1/2*1/(a^4*x^2) - Log[a - x]/(4*a^6) - Log[a + x]/(4*a^6) + Log[a^2 + x^2]/(4*a^6)

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Maple [A]
time = 0.05, size = 43, normalized size = 1.65

method	result	size
risch	\(-\frac {1}{2 a^{4} x^{2}}+\frac {\ln \left (-a^{2}-x^{2}\right )}{4 a^{6}}-\frac {\ln \left (a^{2}-x^{2}\right )}{4 a^{6}}\)	\(42\)
default	\(\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{6}}-\frac {1}{2 a^{4} x^{2}}-\frac {\ln \left (a +x \right )}{4 a^{6}}-\frac {\ln \left (a -x \right )}{4 a^{6}}\)	\(43\)
norman	\(\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{6}}-\frac {1}{2 a^{4} x^{2}}-\frac {\ln \left (a +x \right )}{4 a^{6}}-\frac {\ln \left (a -x \right )}{4 a^{6}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a^6*ln(a^2+x^2)-1/2/a^4/x^2-1/4*ln(a+x)/a^6-1/4/a^6*ln(a-x)

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Maxima [A]
time = 1.73, size = 37, normalized size = 1.42 \begin {gather*} \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac {\log \left (-a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac {1}{2 \, a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(a^2 + x^2)/a^6 - 1/4*log(-a^2 + x^2)/a^6 - 1/2/(a^4*x^2)

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Fricas [A]
time = 0.39, size = 41, normalized size = 1.58 \begin {gather*} \frac {x^{2} \log \left (a^{2} + x^{2}\right ) - x^{2} \log \left (-a^{2} + x^{2}\right ) - 2 \, a^{2}}{4 \, a^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^2*log(a^2 + x^2) - x^2*log(-a^2 + x^2) - 2*a^2)/(a^6*x^2)

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Sympy [A]
time = 0.09, size = 34, normalized size = 1.31 \begin {gather*} - \frac {1}{2 a^{4} x^{2}} - \frac {\frac {\log {\left (- a^{2} + x^{2} \right )}}{4} - \frac {\log {\left (a^{2} + x^{2} \right )}}{4}}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a**4*x**2) - (log(-a**2 + x**2)/4 - log(a**2 + x**2)/4)/a**6

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Giac [A]
time = 1.14, size = 38, normalized size = 1.46 \begin {gather*} \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac {\log \left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{6}} - \frac {1}{2 \, a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(a^2 + x^2)/a^6 - 1/4*log(abs(-a^2 + x^2))/a^6 - 1/2/(a^4*x^2)

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Mupad [B]
time = 0.20, size = 22, normalized size = 0.85 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {x^2}{a^2}\right )}{2\,a^6}-\frac {1}{2\,a^4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a^4 - x^4)),x)

[Out]

atanh(x^2/a^2)/(2*a^6) - 1/(2*a^4*x^2)

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