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

Optimal result

Integrand size = 22, antiderivative size = 26 \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a (1+a x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {665} \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a (a x+1)} \]

[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*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2}}{a (1+a x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a (1+a x)} \]

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/(a*(1 + a*x)))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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	\(-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{2} \left (x +\frac {1}{a}\right )}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} x + a} \]

[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)

Sympy [F]

\[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2} x + a} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {2}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]

[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))

Mupad [B] (verification not implemented)

Time = 11.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{x\,a^2+a} \]

[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)