3.2.53 \(\int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx\) [153]

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

[Out]

-(-a^2*x^2+1)^(1/2)/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 = {665} \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 665

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 = 0.12, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a (1+a x)} \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*(1 + a*x)))

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Maple [A]
time = 0.06, size = 36, normalized size = 1.38

method	result	size
gosper	\(\frac {a x -1}{a \sqrt {-a^{2} x^{2}+1}}\)	\(22\)
trager	\(-\frac {\sqrt {-a^{2} x^{2}+1}}{a \left (a x +1\right )}\)	\(25\)
default	\(-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{2} \left (x +\frac {1}{a}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, 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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Fricas [A]
time = 2.24, 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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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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Giac [A]
time = 0.93, 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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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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