3.337 \(\int \frac{\sqrt{x}}{1-x^2} \, dx\)

Optimal. Leaf size=15 \[ \tanh ^{-1}\left (\sqrt{x}\right )-\tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

-ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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Rubi [A]  time = 0.0074245, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {329, 298, 203, 206} \[ \tanh ^{-1}\left (\sqrt{x}\right )-\tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 - x^2),x]

[Out]

-ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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 298

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

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])

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{1-x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{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 = 0.0036275, size = 15, normalized size = 1. \[ \tanh ^{-1}\left (\sqrt{x}\right )-\tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 - x^2),x]

[Out]

-ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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Maple [B]  time = 0.007, size = 24, normalized size = 1.6 \begin{align*} -{\frac{1}{2}\ln \left ( -1+\sqrt{x} \right ) }+{\frac{1}{2}\ln \left ( \sqrt{x}+1 \right ) }-\arctan \left ( \sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*ln(-1+x^(1/2))+1/2*ln(x^(1/2)+1)-arctan(x^(1/2))

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Maxima [B]  time = 2.09145, size = 31, normalized size = 2.07 \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^2+1),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 = 1.66467, size = 86, normalized size = 5.73 \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^2+1),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 = 0.299491, size = 26, normalized size = 1.73 \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**2+1),x)

[Out]

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

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Giac [B]  time = 2.82384, size = 32, normalized size = 2.13 \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^2+1),x, algorithm="giac")

[Out]

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