3.1316 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{x (c-a^2 c x^2)} \, dx\)

Optimal. Leaf size=90 \[ \frac{(1-a x)^{-n/2} (a x+1)^{n/2}}{c n}-\frac{2 (1-a x)^{-n/2} (a x+1)^{n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a x+1}{1-a x}\right )}{c n} \]

[Out]

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

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Rubi [A]  time = 0.105493, antiderivative size = 100, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6150, 96, 131} \[ \frac{(1-a x)^{-n/2} (a x+1)^{n/2}}{c n}-\frac{2 (1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n-2}{2}} \, _2F_1\left (1,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{c (2-n)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 6150

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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

Mathematica [A]  time = 0.0352695, size = 85, normalized size = 0.94 \[ \frac{(1-a x)^{-n/2} (a x+1)^{\frac{n}{2}-1} \left ((n-2) (a x+1)-2 n (a x-1) \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )\right )}{c (n-2) n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{x \left ( -{a}^{2}c{x}^{2}+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(n*arctanh(a*x))/x/(-a^2*c*x^2+c),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}}{{\left (a^{2} c x^{2} - c\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/((a^2*c*x^2 - c)*x), 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}}{a^{2} c x^{3} - c x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-((a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^3 - c*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{2} x^{3} - x}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(exp(n*atanh(a*x))/(a**2*x**3 - x), x)/c

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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}}{{\left (a^{2} c x^{2} - c\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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