3.8.67 \(\int \frac {1}{\sqrt {1-x} (1+x)} \, dx\)

Optimal. Leaf size=23 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {63, 206} \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*(1 + x)),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x]*(1 + x)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[1 - x]*(1 + x)),x]

[Out]

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

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fricas [A]  time = 0.39, size = 27, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x + 2 \, \sqrt {2} \sqrt {-x + 1} - 3}{x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 38, normalized size = 1.65 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x + 1}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {2} + \sqrt {-x + 1} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 19, normalized size = 0.83 \begin {gather*} -\sqrt {2}\, \arctanh \left (\frac {\sqrt {-x +1}\, \sqrt {2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 34, normalized size = 1.48 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {2} + \sqrt {-x + 1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 18, normalized size = 0.78 \begin {gather*} -\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {1-x}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(1/2)*(x + 1)),x)

[Out]

-2^(1/2)*atanh((2^(1/2)*(1 - x)^(1/2))/2)

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sympy [A]  time = 1.02, size = 44, normalized size = 1.91 \begin {gather*} \begin {cases} - \sqrt {2} \operatorname {acosh}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\\sqrt {2} i \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(2)*acosh(sqrt(2)/sqrt(x + 1)), 2/Abs(x + 1) > 1), (sqrt(2)*I*asin(sqrt(2)/sqrt(x + 1)), True)
)

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