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

Optimal. Leaf size=67 \[ -\frac{4 a (1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n-2}{2}} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )}{2-n} \]

[Out]

(-4*a*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/
(2 - n)

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Rubi [A]  time = 0.0311097, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 131} \[ -\frac{4 a (1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{2-n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/
(2 - n)

Rule 6126

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

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{x^2} \, dx\\ &=-\frac{4 a (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{1+a x}\right )}{2-n}\\ \end{align*}

Mathematica [A]  time = 0.0163086, size = 65, normalized size = 0.97 \[ \frac{4 a (1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n}{2}-1} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )}{n-2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*(1 - a*x)^(1 - n/2)*(1 + a*x)^(-1 + n/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/(-2
+ n)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(n*atanh(a*x))/x**2, x)

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Giac [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}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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