3.1511 \(\int \frac{\sqrt{1+x^8}}{x} \, dx\)

Optimal. Leaf size=28 \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

[Out]

Sqrt[1 + x^8]/4 - ArcTanh[Sqrt[1 + x^8]]/4

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Rubi [A]  time = 0.0109673, antiderivative size = 28, 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, 50, 63, 207} \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^8]/x,x]

[Out]

Sqrt[1 + x^8]/4 - 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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.0039733, size = 28, normalized size = 1. \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^8]/x,x]

[Out]

Sqrt[1 + x^8]/4 - ArcTanh[Sqrt[1 + x^8]]/4

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Maple [B]  time = 0.034, size = 56, normalized size = 2. \begin{align*} -{\frac{1}{16\,\sqrt{\pi }} \left ( -2\, \left ( 2-2\,\ln \left ( 2 \right ) +8\,\ln \left ( x \right ) \right ) \sqrt{\pi }+4\,\sqrt{\pi }-4\,\sqrt{\pi }\sqrt{{x}^{8}+1}+4\,\sqrt{\pi }\ln \left ( 1/2+1/2\,\sqrt{{x}^{8}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16/Pi^(1/2)*(-2*(2-2*ln(2)+8*ln(x))*Pi^(1/2)+4*Pi^(1/2)-4*Pi^(1/2)*(x^8+1)^(1/2)+4*Pi^(1/2)*ln(1/2+1/2*(x^8
+1)^(1/2)))

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Maxima [A]  time = 0.968956, size = 46, normalized size = 1.64 \begin{align*} \frac{1}{4} \, \sqrt{x^{8} + 1} - \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((x^8+1)^(1/2)/x,x, algorithm="maxima")

[Out]

1/4*sqrt(x^8 + 1) - 1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)

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Fricas [A]  time = 1.30886, size = 104, normalized size = 3.71 \begin{align*} \frac{1}{4} \, \sqrt{x^{8} + 1} - \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((x^8+1)^(1/2)/x,x, algorithm="fricas")

[Out]

1/4*sqrt(x^8 + 1) - 1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)

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Sympy [A]  time = 1.11073, size = 39, normalized size = 1.39 \begin{align*} \frac{x^{4}}{4 \sqrt{1 + \frac{1}{x^{8}}}} - \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{4} + \frac{1}{4 x^{4} \sqrt{1 + \frac{1}{x^{8}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/(4*sqrt(1 + x**(-8))) - asinh(x**(-4))/4 + 1/(4*x**4*sqrt(1 + x**(-8)))

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Giac [A]  time = 1.16355, size = 46, normalized size = 1.64 \begin{align*} \frac{1}{4} \, \sqrt{x^{8} + 1} - \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((x^8+1)^(1/2)/x,x, algorithm="giac")

[Out]

1/4*sqrt(x^8 + 1) - 1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)