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

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 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])

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])

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*}

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Mathematica [A]  time = 0.01, 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)

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fricas [A]  time = 0.58, size = 40, normalized size = 1.48 \[ \frac {{\left (a x - 1\right )} \log \left (a x + 1\right ) - {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \, {\left (a^{2} x - a\right )}} \]

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)

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giac [A]  time = 0.49, size = 37, normalized size = 1.37 \[ \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{4 \, a} - \frac {1}{2 \, {\left (a x - 1\right )} a} \]

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)

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maple [A]  time = 0.03, size = 36, normalized size = 1.33 \[ -\frac {1}{2 a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{4 a}+\frac {\ln \left (a x +1\right )}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 36, normalized size = 1.33 \[ \frac {\log \left (a x + 1\right )}{4 \, a} - \frac {\log \left (a x - 1\right )}{4 \, a} - \frac {1}{2 \, {\left (a^{2} x - a\right )}} \]

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)

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mupad [B]  time = 0.92, size = 22, normalized size = 0.81 \[ \frac {\mathrm {atanh}\left (a\,x\right )}{2\,a}-\frac {1}{2\,a\,\left (a\,x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)/(a^2*x^2 - 1)^2,x)

[Out]

atanh(a*x)/(2*a) - 1/(2*a*(a*x - 1))

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sympy [A]  time = 0.18, size = 29, normalized size = 1.07 \[ - \frac {1}{2 a^{2} x - 2 a} + \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{4} + \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a} \]

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

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