### 3.928 $$\int \frac{1}{\sqrt{1-a x} (1+a x)} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0081858, antiderivative size = 27, normalized size of antiderivative = 1., 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} $-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a}$

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*}

Mathematica [A]  time = 0.0069099, size = 27, normalized size = 1. $-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a}$

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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Maple [A]  time = 0.043, size = 23, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{a}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ax+1}} \right ) } \end{align*}

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 = 1.67271, size = 53, normalized size = 1.96 \begin{align*} \frac{\sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{-a x + 1}}{\sqrt{2} + \sqrt{-a x + 1}}\right )}{2 \, a} \end{align*}

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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Fricas [A]  time = 1.78686, size = 90, normalized size = 3.33 \begin{align*} \frac{\sqrt{2} \log \left (\frac{a x + 2 \, \sqrt{2} \sqrt{-a x + 1} - 3}{a x + 1}\right )}{2 \, a} \end{align*}

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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Sympy [A]  time = 5.7172, size = 65, normalized size = 2.41 \begin{align*} \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{align*}

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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Giac [A]  time = 1.14819, size = 57, normalized size = 2.11 \begin{align*} \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{align*}

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