3.19 \(\int \frac{\sqrt{1+x^3}}{x} \, dx\)

Optimal. Leaf size=28 \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

[Out]

(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3

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Rubi [A]  time = 0.0094395, 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} \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^3]/x,x]

[Out]

(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3

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^3}}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x} \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^3}\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{1+x^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0054956, size = 28, normalized size = 1. \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^3]/x,x]

[Out]

(2*Sqrt[1 + x^3])/3 - (2*ArcTanh[Sqrt[1 + x^3]])/3

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Maple [A]  time = 0.017, size = 21, normalized size = 0.8 \begin{align*} -{\frac{2}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }+{\frac{2}{3}\sqrt{{x}^{3}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(1/2)/x,x)

[Out]

-2/3*arctanh((x^3+1)^(1/2))+2/3*(x^3+1)^(1/2)

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Maxima [A]  time = 0.953164, size = 46, normalized size = 1.64 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(1/2)/x,x, algorithm="maxima")

[Out]

2/3*sqrt(x^3 + 1) - 1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1)

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Fricas [A]  time = 1.88637, size = 104, normalized size = 3.71 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(1/2)/x,x, algorithm="fricas")

[Out]

2/3*sqrt(x^3 + 1) - 1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1)

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Sympy [A]  time = 1.12455, size = 48, normalized size = 1.71 \begin{align*} \frac{2 x^{\frac{3}{2}}}{3 \sqrt{1 + \frac{1}{x^{3}}}} - \frac{2 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{2}{3 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(1/2)/x,x)

[Out]

2*x**(3/2)/(3*sqrt(1 + x**(-3))) - 2*asinh(x**(-3/2))/3 + 2/(3*x**(3/2)*sqrt(1 + x**(-3)))

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Giac [A]  time = 1.09201, size = 47, normalized size = 1.68 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} - \frac{1}{3} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(1/2)/x,x, algorithm="giac")

[Out]

2/3*sqrt(x^3 + 1) - 1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(abs(sqrt(x^3 + 1) - 1))