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

Optimal. Leaf size=27 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a} \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {63, 206} \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - a*x]*(1 + a*x)),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 27, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 35, normalized size = 1.30 \begin {gather*} \frac {\sqrt {2} \log \left (\frac {a x + 2 \, \sqrt {2} \sqrt {-a x + 1} - 3}{a x + 1}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 42, normalized size = 1.56 \begin {gather*} \frac {\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {-a x + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {-a x + 1}\right )}}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {-a x +1}\, \sqrt {2}}{2}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.86, size = 39, normalized size = 1.44 \begin {gather*} \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {-a x + 1}}{\sqrt {2} + \sqrt {-a x + 1}}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.47, size = 19, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2-2\,a\,x}}{2}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - a*x)^(1/2)*(a*x + 1)),x)

[Out]

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

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sympy [A]  time = 6.18, size = 65, normalized size = 2.41 \begin {gather*} \begin {cases} \frac {2 \left (\begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2}}{\sqrt {- a x + 1}} \right )}}{2} & \text {for}\: \frac {1}{- a x + 1} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2}}{\sqrt {- a x + 1}} \right )}}{2} & \text {for}\: \frac {1}{- a x + 1} < \frac {1}{2} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*Piecewise((-sqrt(2)*acoth(sqrt(2)/sqrt(-a*x + 1))/2, 1/(-a*x + 1) > 1/2), (-sqrt(2)*atanh(sqrt(2)
/sqrt(-a*x + 1))/2, 1/(-a*x + 1) < 1/2))/a, Ne(a, 0)), (x, True))

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