3.958 \(\int \frac{\sqrt{x}}{x-x^3} \, dx\)

Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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Rubi [A]  time = 0.0106321, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1584, 329, 212, 206, 203} \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(x - x^3),x]

[Out]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{x-x^3} \, dx &=\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{x}\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0032254, size = 13, normalized size = 1. \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(x - x^3),x]

[Out]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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Maple [A]  time = 0.005, size = 10, normalized size = 0.8 \begin{align*} \arctan \left ( \sqrt{x} \right ) +{\it Artanh} \left ( \sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x^(1/2))+arctanh(x^(1/2))

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Maxima [B]  time = 1.66253, size = 28, normalized size = 2.15 \begin{align*} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x)) + 1/2*log(sqrt(x) + 1) - 1/2*log(sqrt(x) - 1)

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Fricas [B]  time = 1.46895, size = 85, normalized size = 6.54 \begin{align*} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x)) + 1/2*log(sqrt(x) + 1) - 1/2*log(sqrt(x) - 1)

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Sympy [B]  time = 0.573867, size = 26, normalized size = 2. \begin{align*} - \frac{\log{\left (\sqrt{x} - 1 \right )}}{2} + \frac{\log{\left (\sqrt{x} + 1 \right )}}{2} + \operatorname{atan}{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(x) - 1)/2 + log(sqrt(x) + 1)/2 + atan(sqrt(x))

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Giac [B]  time = 1.16499, size = 30, normalized size = 2.31 \begin{align*} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left ({\left | \sqrt{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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