3.130 \(\int \frac{1}{x^2 (a^4-x^4)} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{a^4 x}-\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^5}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^5} \]

[Out]

-(1/(a^4*x)) - ArcTan[x/a]/(2*a^5) + ArcTanh[x/a]/(2*a^5)

________________________________________________________________________________________

Rubi [A]  time = 0.013661, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {325, 298, 203, 206} \[ -\frac{1}{a^4 x}-\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^5}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a^4 - x^4)),x]

[Out]

-(1/(a^4*x)) - ArcTan[x/a]/(2*a^5) + ArcTanh[x/a]/(2*a^5)

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 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

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^2 \left (a^4-x^4\right )} \, dx &=-\frac{1}{a^4 x}+\frac{\int \frac{x^2}{a^4-x^4} \, dx}{a^4}\\ &=-\frac{1}{a^4 x}+\frac{\int \frac{1}{a^2-x^2} \, dx}{2 a^4}-\frac{\int \frac{1}{a^2+x^2} \, dx}{2 a^4}\\ &=-\frac{1}{a^4 x}-\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^5}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0067181, size = 46, normalized size = 1.31 \[ -\frac{1}{a^4 x}-\frac{\log (a-x)}{4 a^5}+\frac{\log (a+x)}{4 a^5}-\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a^4 - x^4)),x]

[Out]

-(1/(a^4*x)) - ArcTan[x/a]/(2*a^5) - Log[a - x]/(4*a^5) + Log[a + x]/(4*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 41, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{5}}}-{\frac{1}{2\,{a}^{5}}\arctan \left ({\frac{x}{a}} \right ) }+{\frac{\ln \left ( a+x \right ) }{4\,{a}^{5}}}-{\frac{1}{{a}^{4}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a^4-x^4),x)

[Out]

-1/4/a^5*ln(-a+x)-1/2*arctan(x/a)/a^5+1/4*ln(a+x)/a^5-1/a^4/x

________________________________________________________________________________________

Maxima [A]  time = 1.41269, size = 54, normalized size = 1.54 \begin{align*} -\frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{5}} + \frac{\log \left (a + x\right )}{4 \, a^{5}} - \frac{\log \left (-a + x\right )}{4 \, a^{5}} - \frac{1}{a^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^4-x^4),x, algorithm="maxima")

[Out]

-1/2*arctan(x/a)/a^5 + 1/4*log(a + x)/a^5 - 1/4*log(-a + x)/a^5 - 1/(a^4*x)

________________________________________________________________________________________

Fricas [A]  time = 1.97679, size = 93, normalized size = 2.66 \begin{align*} -\frac{2 \, x \arctan \left (\frac{x}{a}\right ) - x \log \left (a + x\right ) + x \log \left (-a + x\right ) + 4 \, a}{4 \, a^{5} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^4-x^4),x, algorithm="fricas")

[Out]

-1/4*(2*x*arctan(x/a) - x*log(a + x) + x*log(-a + x) + 4*a)/(a^5*x)

________________________________________________________________________________________

Sympy [C]  time = 0.329665, size = 44, normalized size = 1.26 \begin{align*} - \frac{1}{a^{4} x} - \frac{\frac{\log{\left (- a + x \right )}}{4} - \frac{\log{\left (a + x \right )}}{4} - \frac{i \log{\left (- i a + x \right )}}{4} + \frac{i \log{\left (i a + x \right )}}{4}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a**4-x**4),x)

[Out]

-1/(a**4*x) - (log(-a + x)/4 - log(a + x)/4 - I*log(-I*a + x)/4 + I*log(I*a + x)/4)/a**5

________________________________________________________________________________________

Giac [A]  time = 1.06125, size = 57, normalized size = 1.63 \begin{align*} -\frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{5}} + \frac{\log \left ({\left | a + x \right |}\right )}{4 \, a^{5}} - \frac{\log \left ({\left | -a + x \right |}\right )}{4 \, a^{5}} - \frac{1}{a^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^4-x^4),x, algorithm="giac")

[Out]

-1/2*arctan(x/a)/a^5 + 1/4*log(abs(a + x))/a^5 - 1/4*log(abs(-a + x))/a^5 - 1/(a^4*x)