3.3 \(\int \frac{1}{x \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

[Out]

-ArcTanh[Sqrt[1 + x^8]]/4

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Rubi [A]  time = 0.0063612, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 63, 207} \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 + x^8]),x]

[Out]

-ArcTanh[Sqrt[1 + x^8]]/4

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{1+x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^8\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^8}\right )\\ &=-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1+x^8}\right )\\ \end{align*}

Mathematica [A]  time = 0.0023587, size = 14, normalized size = 1. \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 + x^8]),x]

[Out]

-ArcTanh[Sqrt[1 + x^8]]/4

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Maple [A]  time = 0., size = 19, normalized size = 1.4 \begin{align*}{\frac{1}{4}\ln \left ({ \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8+1)^(1/2)/x,x)

[Out]

1/4*ln(((x^8+1)^(1/2)-1)/(x^8)^(1/2))

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Maxima [B]  time = 0.959928, size = 34, normalized size = 2.43 \begin{align*} -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8+1)^(1/2)/x,x, algorithm="maxima")

[Out]

-1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)

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Fricas [B]  time = 2.19793, size = 78, normalized size = 5.57 \begin{align*} -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8+1)^(1/2)/x,x, algorithm="fricas")

[Out]

-1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)

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Sympy [A]  time = 0.956568, size = 8, normalized size = 0.57 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**8+1)**(1/2)/x,x)

[Out]

-asinh(x**(-4))/4

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Giac [B]  time = 1.07175, size = 34, normalized size = 2.43 \begin{align*} -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8+1)^(1/2)/x,x, algorithm="giac")

[Out]

-1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)