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

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \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]*Sqrt[1 - a^2*x^2]),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])

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]))

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*}

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Mathematica [A]  time = 0.05, size = 53, normalized size = 1.96 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.10, size = 47, normalized size = 1.74 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a x+1}}{\sqrt {2 (a x+1)-(a x+1)^2}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 63, normalized size = 2.33 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.16, size = 43, normalized size = 1.59 \begin {gather*} -\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 {gather*}

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

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maple [B]  time = 0.09, size = 50, normalized size = 1.85 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a x +1}\, \sqrt {2}}{2}\right )}{\sqrt {a x +1}\, \sqrt {-a x +1}\, a} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {1-a^2\,x^2}\,\sqrt {a\,x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {a x + 1}}\, dx \end {gather*}

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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