3.933 \(\int \frac{e^{\tanh ^{-1}(a x)}}{(1-a^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0484609, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6140, 44, 207} \[ \frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(1 - a^2*x^2)^(3/2),x]

[Out]

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

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

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

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{2 (-1+a x)^2}-\frac{1}{2 \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{2 a (1-a x)}-\frac{1}{2} \int \frac{1}{-1+a^2 x^2} \, dx\\ &=\frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.015646, size = 20, normalized size = 0.74 \[ \frac{\frac{1}{1-a x}+\tanh ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(1 - a^2*x^2)^(3/2),x]

[Out]

((1 - a*x)^(-1) + ArcTanh[a*x])/(2*a)

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 36, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( ax+1 \right ) }{4\,a}}-{\frac{1}{2\,a \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(a*x+1)/a-1/2/a/(a*x-1)-1/4/a*ln(a*x-1)

________________________________________________________________________________________

Maxima [A]  time = 0.94299, size = 49, normalized size = 1.81 \begin{align*} \frac{\log \left (a x + 1\right )}{4 \, a} - \frac{\log \left (a x - 1\right )}{4 \, a} - \frac{1}{2 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.69701, size = 96, normalized size = 3.56 \begin{align*} \frac{{\left (a x - 1\right )} \log \left (a x + 1\right ) -{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.352961, size = 29, normalized size = 1.07 \begin{align*} - \frac{1}{2 a^{2} x - 2 a} + \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{4} + \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16653, size = 50, normalized size = 1.85 \begin{align*} \frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a} - \frac{\log \left ({\left | a x - 1 \right |}\right )}{4 \, a} - \frac{1}{2 \,{\left (a x - 1\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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