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

Optimal. Leaf size=24 \[ \frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4} \]

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {266, 36, 31, 29} \[ \frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a^4 - x^4)),x]

[Out]

Log[x]/a^4 - Log[a^4 - x^4]/(4*a^4)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a^4-x^4\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\left (a^4-x\right ) x} \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a^4-x} \, dx,x,x^4\right )}{4 a^4}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )}{4 a^4}\\ &=\frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \[ \frac {\log (x)}{a^4}-\frac {\log \left (x^4-a^4\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a^4 - x^4)),x]

[Out]

Log[x]/a^4 - Log[-a^4 + x^4]/(4*a^4)

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IntegrateAlgebraic [A]  time = 0.01, size = 24, normalized size = 1.00 \[ \frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(a^4 - x^4)),x]

[Out]

Log[x]/a^4 - Log[a^4 - x^4]/(4*a^4)

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fricas [A]  time = 0.59, size = 20, normalized size = 0.83 \[ -\frac {\log \left (-a^{4} + x^{4}\right ) - 4 \, \log \relax (x)}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(log(-a^4 + x^4) - 4*log(x))/a^4

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giac [A]  time = 0.95, size = 26, normalized size = 1.08 \[ \frac {\log \left (x^{4}\right )}{4 \, a^{4}} - \frac {\log \left ({\left | -a^{4} + x^{4} \right |}\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.32, size = 23, normalized size = 0.96

method	result	size
risch	\(\frac {\ln \relax (x )}{a^{4}}-\frac {\ln \left (-a^{4}+x^{4}\right )}{4 a^{4}}\)	\(23\)
default	\(-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{4}}+\frac {\ln \relax (x )}{a^{4}}-\frac {\ln \left (a +x \right )}{4 a^{4}}-\frac {\ln \left (a -x \right )}{4 a^{4}}\)	\(41\)
norman	\(-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{4}}+\frac {\ln \relax (x )}{a^{4}}-\frac {\ln \left (a +x \right )}{4 a^{4}}-\frac {\ln \left (a -x \right )}{4 a^{4}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 25, normalized size = 1.04 \[ -\frac {\log \left (-a^{4} + x^{4}\right )}{4 \, a^{4}} + \frac {\log \left (x^{4}\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.28, size = 20, normalized size = 0.83 \[ -\frac {\ln \left (x^4-a^4\right )-4\,\ln \relax (x)}{4\,a^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a^4 - x^4)),x)

[Out]

-(log(x^4 - a^4) - 4*log(x))/(4*a^4)

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sympy [A]  time = 0.25, size = 19, normalized size = 0.79 \[ \frac {\log {\relax (x )}}{a^{4}} - \frac {\log {\left (- a^{4} + x^{4} \right )}}{4 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/a**4 - log(-a**4 + x**4)/(4*a**4)

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