3.444 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n+2}{2}}}{a c^2 (n+2)} \]

[Out]

((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^2*(2 + n))

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Rubi [A]  time = 0.0389181, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 37} \[ \frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n+2}{2}}}{a c^2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])/(c - a*c*x)^2,x]

[Out]

((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^2*(2 + n))

Rule 6129

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0191819, size = 39, normalized size = 1. \[ \frac{(1-a x)^{-\frac{n}{2}-1} (a x+1)^{\frac{n}{2}+1}}{a c^2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTanh[a*x])/(c - a*c*x)^2,x]

[Out]

((1 - a*x)^(-1 - n/2)*(1 + a*x)^(1 + n/2))/(a*c^2*(2 + n))

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Maple [A]  time = 0.033, size = 33, normalized size = 0.9 \begin{align*} -{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( ax+1 \right ) }{ \left ( ax-1 \right ){c}^{2} \left ( 2+n \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/(-a*c*x+c)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a c x - c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92778, size = 120, normalized size = 3.08 \begin{align*} \frac{{\left (a x + 1\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c^{2} n + 2 \, a c^{2} -{\left (a^{2} c^{2} n + 2 \, a^{2} c^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

(a*x + 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^2*n + 2*a*c^2 - (a^2*c^2*n + 2*a^2*c^2)*x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/(-a*c*x+c)**2,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.11933, size = 43, normalized size = 1.1 \begin{align*} -\frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n + 1}}{a c^{2}{\left (n + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^2,x, algorithm="giac")

[Out]

-((a*x + 1)/(a*x - 1))^(1/2*n + 1)/(a*c^2*(n + 2))