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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 13, 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 = {329, 212, 206, 203} \[ \tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(1 - x^2)),x]

[Out]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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])

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 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]

Rubi steps

\begin {align*} \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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(1 - x^2)),x]

[Out]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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fricas [B]  time = 0.57, size = 21, normalized size = 1.62 \[ \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.39, size = 22, normalized size = 1.69 \[ \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 10, normalized size = 0.77 \[ \arctanh \left (\sqrt {x}\right )+\arctan \left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.01, size = 21, normalized size = 1.62 \[ \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.34, size = 9, normalized size = 0.69 \[ \mathrm {atan}\left (\sqrt {x}\right )+\mathrm {atanh}\left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^(1/2)*(x^2 - 1)),x)

[Out]

atan(x^(1/2)) + atanh(x^(1/2))

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sympy [B]  time = 0.39, size = 26, normalized size = 2.00 \[ - \frac {\log {\left (\sqrt {x} - 1 \right )}}{2} + \frac {\log {\left (\sqrt {x} + 1 \right )}}{2} + \operatorname {atan}{\left (\sqrt {x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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