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

Optimal. Leaf size=37 \[ -\frac{1}{3 a^4 x^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7} \]

[Out]

-1/(3*a^4*x^3) + ArcTan[x/a]/(2*a^7) + ArcTanh[x/a]/(2*a^7)

________________________________________________________________________________________

Rubi [A]  time = 0.0122327, antiderivative size = 37, 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, 212, 206, 203} \[ -\frac{1}{3 a^4 x^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a^4 - x^4)),x]

[Out]

-1/(3*a^4*x^3) + ArcTan[x/a]/(2*a^7) + ArcTanh[x/a]/(2*a^7)

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 212

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

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])

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a^4-x^4\right )} \, dx &=-\frac{1}{3 a^4 x^3}+\frac{\int \frac{1}{a^4-x^4} \, dx}{a^4}\\ &=-\frac{1}{3 a^4 x^3}+\frac{\int \frac{1}{a^2-x^2} \, dx}{2 a^6}+\frac{\int \frac{1}{a^2+x^2} \, dx}{2 a^6}\\ &=-\frac{1}{3 a^4 x^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7}\\ \end{align*}

Mathematica [A]  time = 0.0068022, size = 48, normalized size = 1.3 \[ -\frac{1}{3 a^4 x^3}-\frac{\log (a-x)}{4 a^7}+\frac{\log (a+x)}{4 a^7}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a^4 - x^4)),x]

[Out]

-1/(3*a^4*x^3) + ArcTan[x/a]/(2*a^7) - Log[a - x]/(4*a^7) + Log[a + x]/(4*a^7)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 41, normalized size = 1.1 \begin{align*} -{\frac{1}{3\,{a}^{4}{x}^{3}}}-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{7}}}+{\frac{1}{2\,{a}^{7}}\arctan \left ({\frac{x}{a}} \right ) }+{\frac{\ln \left ( a+x \right ) }{4\,{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/a^4/x^3-1/4/a^7*ln(-a+x)+1/2*arctan(x/a)/a^7+1/4*ln(a+x)/a^7

________________________________________________________________________________________

Maxima [A]  time = 1.40574, size = 54, normalized size = 1.46 \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{7}} + \frac{\log \left (a + x\right )}{4 \, a^{7}} - \frac{\log \left (-a + x\right )}{4 \, a^{7}} - \frac{1}{3 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x/a)/a^7 + 1/4*log(a + x)/a^7 - 1/4*log(-a + x)/a^7 - 1/3/(a^4*x^3)

________________________________________________________________________________________

Fricas [A]  time = 1.84476, size = 112, normalized size = 3.03 \begin{align*} \frac{6 \, x^{3} \arctan \left (\frac{x}{a}\right ) + 3 \, x^{3} \log \left (a + x\right ) - 3 \, x^{3} \log \left (-a + x\right ) - 4 \, a^{3}}{12 \, a^{7} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*x^3*arctan(x/a) + 3*x^3*log(a + x) - 3*x^3*log(-a + x) - 4*a^3)/(a^7*x^3)

________________________________________________________________________________________

Sympy [C]  time = 0.368821, size = 48, normalized size = 1.3 \begin{align*} - \frac{1}{3 a^{4} x^{3}} - \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^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(3*a**4*x**3) - (log(-a + x)/4 - log(a + x)/4 + I*log(-I*a + x)/4 - I*log(I*a + x)/4)/a**7

________________________________________________________________________________________

Giac [A]  time = 1.04898, size = 57, normalized size = 1.54 \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{7}} + \frac{\log \left ({\left | a + x \right |}\right )}{4 \, a^{7}} - \frac{\log \left ({\left | -a + x \right |}\right )}{4 \, a^{7}} - \frac{1}{3 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x/a)/a^7 + 1/4*log(abs(a + x))/a^7 - 1/4*log(abs(-a + x))/a^7 - 1/3/(a^4*x^3)