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

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a (a x+1)} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a^2 x+a} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.32, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a (a x+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/(a*(1 + a*x)))

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fricas [A]  time = 0.39, size = 28, normalized size = 1.08 \begin {gather*} -\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} x + a} \end {gather*}

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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giac [A]  time = 0.20, size = 34, normalized size = 1.31 \begin {gather*} \frac {2}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \end {gather*}

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))

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maple [A]  time = 0.01, size = 22, normalized size = 0.85 \begin {gather*} \frac {a x -1}{\sqrt {-a^{2} x^{2}+1}\, a} \end {gather*}

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 = 0.97, size = 23, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2} x + a} \end {gather*}

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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mupad [B]  time = 2.59, size = 23, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}}{x\,a^2+a} \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)),x)

[Out]

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

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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