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

Optimal. Leaf size=26 \[ -\frac{\sqrt{1-a^2 x^2}}{a (a x+1)} \]

[Out]

-(Sqrt[1 - a^2*x^2]/(a*(1 + a*x)))

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Rubi [A]  time = 0.0095005, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {651} \[ -\frac{\sqrt{1-a^2 x^2}}{a (a x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/(a*(1 + a*x)))

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(1+a x) \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2}}{a (1+a x)}\\ \end{align*}

Mathematica [A]  time = 0.0061963, size = 25, normalized size = 0.96 \[ -\frac{\sqrt{1-a^2 x^2}}{a^2 x+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/(a + a^2*x))

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Maple [A]  time = 0.044, size = 22, normalized size = 0.9 \begin{align*}{\frac{ax-1}{a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47253, size = 31, normalized size = 1.19 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}}{a^{2} x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a^2*x^2 + 1)/(a^2*x + a)

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Fricas [A]  time = 1.54348, size = 61, normalized size = 2.35 \begin{align*} -\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 1}{a^{2} x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x + sqrt(-a^2*x^2 + 1) + 1)/(a^2*x + 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 )} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.25429, size = 46, normalized size = 1.77 \begin{align*} \frac{2}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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