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

Optimal. Leaf size=30 \[ -\frac{\log ^2\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0232066, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {2505} \[ -\frac{\log ^2\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2505

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(
b*c - a*d)), x] /; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && NeQ[s, -1]

Rubi steps

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

Mathematica [A]  time = 0.0085736, size = 30, normalized size = 1. \[ -\frac{\log ^2\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.389, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1}\ln \left ({\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.9242, size = 112, normalized size = 3.73 \begin{align*} \frac{1}{2} \,{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \log \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right ) + \frac{\log \left (a x - 1\right )^{2}}{8 \, a} + \frac{\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(a*x + 1)/a - log(a*x - 1)/a)*log(sqrt(-a*x + 1)/sqrt(a*x + 1)) + 1/8*log(a*x - 1)^2/a + 1/8*(log(a*x
+ 1)^2 - 2*log(a*x + 1)*log(a*x - 1))/a

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Fricas [A]  time = 1.94668, size = 59, normalized size = 1.97 \begin{align*} -\frac{\log \left (\frac{\sqrt{-a x + 1}}{\sqrt{a x + 1}}\right )^{2}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 13.5192, size = 65, normalized size = 2.17 \begin{align*} - \frac{\operatorname{atan}^{2}{\left (\frac{x}{\sqrt{- \frac{1}{a^{2}}}} \right )}}{2 a} - \frac{\log{\left (\frac{\sqrt{- a x + 1}}{\sqrt{a x + 1}} \right )} \operatorname{atan}{\left (\frac{x}{\sqrt{- \frac{1}{a^{2}}}} \right )}}{a^{2} \sqrt{- \frac{1}{a^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-atan(x/sqrt(-1/a**2))**2/(2*a) - log(sqrt(-a*x + 1)/sqrt(a*x + 1))*atan(x/sqrt(-1/a**2))/(a**2*sqrt(-1/a**2))

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Giac [B]  time = 1.29754, size = 78, normalized size = 2.6 \begin{align*} \frac{1}{4} \,{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \log \left (-a x + 1\right ) - \frac{\log \left (a x + 1\right )^{2}}{8 \, a} + \frac{\log \left (a x - 1\right )^{2}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(a*x + 1)/a - log(a*x - 1)/a)*log(-a*x + 1) - 1/8*log(a*x + 1)^2/a + 1/8*log(a*x - 1)^2/a