3.29 \(\int e^{4 \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=27 \[ \frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x \]

[Out]

x + 4/(a*(1 - a*x)) + (4*Log[1 - a*x])/a

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Rubi [A]  time = 0.0143055, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6125, 43} \[ \frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x]),x]

[Out]

x + 4/(a*(1 - a*x)) + (4*Log[1 - a*x])/a

Rule 6125

Int[E^(ArcTanh[(a_.)*(x_)]*(n_)), x_Symbol] :> Int[(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, n}, x] &&
!IntegerQ[(n - 1)/2]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{4 \tanh ^{-1}(a x)} \, dx &=\int \frac{(1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (1+\frac{4}{(-1+a x)^2}+\frac{4}{-1+a x}\right ) \, dx\\ &=x+\frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.016691, size = 26, normalized size = 0.96 \[ -\frac{4}{a (a x-1)}+\frac{4 \log (1-a x)}{a}+x \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x]),x]

[Out]

x - 4/(a*(-1 + a*x)) + (4*Log[1 - a*x])/a

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Maple [A]  time = 0.032, size = 26, normalized size = 1. \begin{align*} x-4\,{\frac{1}{a \left ( ax-1 \right ) }}+4\,{\frac{\ln \left ( ax-1 \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.941763, size = 35, normalized size = 1.3 \begin{align*} x + \frac{4 \, \log \left (a x - 1\right )}{a} - \frac{4}{a^{2} x - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2,x, algorithm="maxima")

[Out]

x + 4*log(a*x - 1)/a - 4/(a^2*x - a)

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Fricas [A]  time = 1.80605, size = 81, normalized size = 3. \begin{align*} \frac{a^{2} x^{2} - a x + 4 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 4}{a^{2} x - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.492423, size = 19, normalized size = 0.7 \begin{align*} x - \frac{4}{a^{2} x - a} + \frac{4 \log{\left (a x - 1 \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2,x)

[Out]

x - 4/(a**2*x - a) + 4*log(a*x - 1)/a

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Giac [A]  time = 1.17458, size = 35, normalized size = 1.3 \begin{align*} x + \frac{4 \, \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac{4}{{\left (a x - 1\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2,x, algorithm="giac")

[Out]

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