3.141 \(\int \frac{1}{x (a^5+x^5)} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0083307, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

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]]

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 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]

Rubi steps

\begin{align*} \int \frac{1}{x \left (a^5+x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x \left (a^5+x\right )} \, dx,x,x^5\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^5\right )}{5 a^5}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^5+x} \, dx,x,x^5\right )}{5 a^5}\\ &=\frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0039255, size = 22, normalized size = 1. \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 49, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ({a}^{4}-{a}^{3}x+{a}^{2}{x}^{2}-a{x}^{3}+{x}^{4} \right ) }{5\,{a}^{5}}}+{\frac{\ln \left ( x \right ) }{{a}^{5}}}-{\frac{\ln \left ( a+x \right ) }{5\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^5+x^5),x)

[Out]

-1/5/a^5*ln(a^4-a^3*x+a^2*x^2-a*x^3+x^4)+ln(x)/a^5-1/5*ln(a+x)/a^5

________________________________________________________________________________________

Maxima [A]  time = 0.927172, size = 31, normalized size = 1.41 \begin{align*} -\frac{\log \left (a^{5} + x^{5}\right )}{5 \, a^{5}} + \frac{\log \left (x^{5}\right )}{5 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^5+x^5),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.89974, size = 51, normalized size = 2.32 \begin{align*} -\frac{\log \left (a^{5} + x^{5}\right ) - 5 \, \log \left (x\right )}{5 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^5+x^5),x, algorithm="fricas")

[Out]

-1/5*(log(a^5 + x^5) - 5*log(x))/a^5

________________________________________________________________________________________

Sympy [A]  time = 0.224743, size = 19, normalized size = 0.86 \begin{align*} \frac{\log{\left (x \right )}}{a^{5}} - \frac{\log{\left (a^{5} + x^{5} \right )}}{5 a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a**5+x**5),x)

[Out]

log(x)/a**5 - log(a**5 + x**5)/(5*a**5)

________________________________________________________________________________________

Giac [A]  time = 1.06726, size = 30, normalized size = 1.36 \begin{align*} -\frac{\log \left ({\left | a^{5} + x^{5} \right |}\right )}{5 \, a^{5}} + \frac{\log \left ({\left | x \right |}\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^5+x^5),x, algorithm="giac")

[Out]

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