3.4.37 \(\int \frac {\sqrt {x}}{1-x^2} \, dx\) [337]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {335, 304, 209, 212} \begin {gather*} \tanh ^{-1}\left (\sqrt {x}\right )-\text {ArcTan}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 - x^2),x]

[Out]

-ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

Rule 209

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

Rule 212

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

Rule 304

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 335

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(23\) vs. \(2(11)=22\).
time = 0.19, size = 24, normalized size = 1.60

method	result	size
derivativedivides	\(-\frac {\ln \left (\sqrt {x}-1\right )}{2}+\frac {\ln \left (\sqrt {x}+1\right )}{2}-\arctan \left (\sqrt {x}\right )\)	\(24\)
default	\(-\frac {\ln \left (\sqrt {x}-1\right )}{2}+\frac {\ln \left (\sqrt {x}+1\right )}{2}-\arctan \left (\sqrt {x}\right )\)	\(24\)
meijerg	\(-\frac {x^{\frac {3}{2}} \left (\ln \left (1-\left (x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{2}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{2}\right )^{\frac {1}{4}}\right )\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}\)	\(40\)
trager	\(\frac {\ln \left (\frac {2 \sqrt {x}+1+x}{x -1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x +1}\right )}{2}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(-x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
time = 0.52, size = 23, normalized size = 1.53 \begin {gather*} -\arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
time = 1.06, size = 23, normalized size = 1.53 \begin {gather*} -\arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.10, size = 26, normalized size = 1.73 \begin {gather*} - \frac {\log {\left (\sqrt {x} - 1 \right )}}{2} + \frac {\log {\left (\sqrt {x} + 1 \right )}}{2} - \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).
time = 1.11, size = 24, normalized size = 1.60 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.03, size = 11, normalized size = 0.73 \begin {gather*} \mathrm {atanh}\left (\sqrt {x}\right )-\mathrm {atan}\left (\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^(1/2)/(x^2 - 1),x)

[Out]

atanh(x^(1/2)) - atan(x^(1/2))

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