3.131 \(\int \frac{1}{x^3 (a^4-x^4)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0121596, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {275, 325, 206} \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^6}-\frac{1}{2 a^4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^4 - x^4)),x]

[Out]

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

Rule 275

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 325

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]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0068284, size = 50, normalized size = 1.92 \[ -\frac{1}{2 a^4 x^2}+\frac{\log \left (a^2+x^2\right )}{4 a^6}-\frac{\log (a-x)}{4 a^6}-\frac{\log (a+x)}{4 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a^4 - x^4)),x]

[Out]

-1/(2*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 = 0.006, size = 43, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,{a}^{4}{x}^{2}}}-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{6}}}+{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{4\,{a}^{6}}}-{\frac{\ln \left ( a+x \right ) }{4\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.924299, size = 50, normalized size = 1.92 \begin{align*} \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{align*}

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 = 1.8199, size = 89, normalized size = 3.42 \begin{align*} \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{align*}

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 = 0.347397, size = 34, normalized size = 1.31 \begin{align*} - \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{align*}

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 = 1.07005, size = 51, normalized size = 1.96 \begin{align*} \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{align*}

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)