\(\int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx\) [929]

Optimal result

Integrand size = 24, antiderivative size = 27 \[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {641, 65, 212} \[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a} \]

[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 65

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
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*} \text {integral}& = \int \frac {1}{\sqrt {1-a x} (1+a x)} \, dx \\ & = -\frac {2 \text {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*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2+2 a x}}{\sqrt {1-a^2 x^2}}\right )}{a} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(22)=44\).

Time = 2.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (\frac {\sqrt {-a x +1}\, \sqrt {2}}{2}\right ) \sqrt {2}}{\sqrt {a x +1}\, \sqrt {-a x +1}\, a}\)	\(50\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {2} \log \left (-\frac {a^{2} x^{2} - 2 \, a x + 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} - 3}{a^{2} x^{2} + 2 \, a x + 1}\right )}{2 \, a} \]

[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

Sympy [F]

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

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {2} \log \left (\sqrt {2} + \sqrt {-a x + 1}\right ) - \sqrt {2} \log \left (\sqrt {2} - \sqrt {-a x + 1}\right )}{2 \, a} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+a x} \sqrt {1-a^2 x^2}} \, dx=\int \frac {1}{\sqrt {1-a^2\,x^2}\,\sqrt {a\,x+1}} \,d x \]

[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)