3.929 \(\int \frac{1}{\sqrt{1+a x} \sqrt{1-a^2 x^2}} \, dx\)

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 = 0.0140314, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {627, 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]*Sqrt[1 - a^2*x^2]),x]

[Out]

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

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} \sqrt{1-a^2 x^2}} \, dx &=\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 = 0.0467998, size = 53, normalized size = 1.96 \[ \frac{\sqrt{a x+1} \sqrt{2 a x-2} \tan ^{-1}\left (\frac{\sqrt{a x-1}}{\sqrt{2}}\right )}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + a*x]*Sqrt[1 - a^2*x^2]),x]

[Out]

(Sqrt[1 + a*x]*Sqrt[-2 + 2*a*x]*ArcTan[Sqrt[-1 + a*x]/Sqrt[2]])/(a*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.122, size = 50, normalized size = 1.9 \begin{align*} -{\frac{\sqrt{2}}{a}\sqrt{-{a}^{2}{x}^{2}+1}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ax+1}} \right ){\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{-ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.77288, size = 149, normalized size = 5.52 \begin{align*} \frac{\sqrt{2} \log \left (-\frac{a^{2} x^{2} - 2 \, a x + 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1} - 3}{a^{2} x^{2} + 2 \, a x + 1}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{a x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(a*x + 1)), x)

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Giac [A]  time = 1.24203, size = 58, normalized size = 2.15 \begin{align*} -\frac{\sqrt{2} \log \left (\sqrt{2} + \sqrt{-a x + 1}\right ) - \sqrt{2} \log \left (\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)^(1/2)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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