3.1340 \(\int \frac{1}{x (a+b x^6)^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (a+b x^6\right )}{6 a^2}+\frac{\log (x)}{a^2}+\frac{1}{6 a \left (a+b x^6\right )} \]

[Out]

1/(6*a*(a + b*x^6)) + Log[x]/a^2 - Log[a + b*x^6]/(6*a^2)

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Rubi [A]  time = 0.0280392, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac{\log \left (a+b x^6\right )}{6 a^2}+\frac{\log (x)}{a^2}+\frac{1}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^6)^2),x]

[Out]

1/(6*a*(a + b*x^6)) + Log[x]/a^2 - Log[a + b*x^6]/(6*a^2)

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \left (a+b x^6\right )^2} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,x^6\right )\\ &=\frac{1}{6 a \left (a+b x^6\right )}+\frac{\log (x)}{a^2}-\frac{\log \left (a+b x^6\right )}{6 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0130811, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+b x^6}-\log \left (a+b x^6\right )+6 \log (x)}{6 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^6)^2),x]

[Out]

(a/(a + b*x^6) + 6*Log[x] - Log[a + b*x^6])/(6*a^2)

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Maple [A]  time = 0.01, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{6\,a \left ( b{x}^{6}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{6}+a \right ) }{6\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^6+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.973108, size = 50, normalized size = 1.32 \begin{align*} \frac{1}{6 \,{\left (a b x^{6} + a^{2}\right )}} - \frac{\log \left (b x^{6} + a\right )}{6 \, a^{2}} + \frac{\log \left (x^{6}\right )}{6 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

1/6/(a*b*x^6 + a^2) - 1/6*log(b*x^6 + a)/a^2 + 1/6*log(x^6)/a^2

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Fricas [A]  time = 1.45866, size = 108, normalized size = 2.84 \begin{align*} -\frac{{\left (b x^{6} + a\right )} \log \left (b x^{6} + a\right ) - 6 \,{\left (b x^{6} + a\right )} \log \left (x\right ) - a}{6 \,{\left (a^{2} b x^{6} + a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

-1/6*((b*x^6 + a)*log(b*x^6 + a) - 6*(b*x^6 + a)*log(x) - a)/(a^2*b*x^6 + a^3)

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Sympy [A]  time = 1.4504, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{6 a^{2} + 6 a b x^{6}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{b} + x^{6} \right )}}{6 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**6+a)**2,x)

[Out]

1/(6*a**2 + 6*a*b*x**6) + log(x)/a**2 - log(a/b + x**6)/(6*a**2)

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Giac [A]  time = 1.17273, size = 63, normalized size = 1.66 \begin{align*} \frac{\log \left (x^{6}\right )}{6 \, a^{2}} - \frac{\log \left ({\left | b x^{6} + a \right |}\right )}{6 \, a^{2}} + \frac{b x^{6} + 2 \, a}{6 \,{\left (b x^{6} + a\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^6+a)^2,x, algorithm="giac")

[Out]

1/6*log(x^6)/a^2 - 1/6*log(abs(b*x^6 + a))/a^2 + 1/6*(b*x^6 + 2*a)/((b*x^6 + a)*a^2)