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

Optimal. Leaf size=66 \[ -\frac{c 2^{\frac{n}{2}+1} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (2-\frac{n}{2},-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (4-n)} \]

[Out]

-((2^(1 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[2 - n/2, -n/2, 3 - n/2, (1 - a*x)/2])/(a*(4 - n)))

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Rubi [A]  time = 0.0376154, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6129, 69} \[ -\frac{c 2^{\frac{n}{2}+1} (1-a x)^{2-\frac{n}{2}} \, _2F_1\left (2-\frac{n}{2},-\frac{n}{2};3-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (4-n)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])*(c - a*c*x),x]

[Out]

-((2^(1 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[2 - n/2, -n/2, 3 - n/2, (1 - a*x)/2])/(a*(4 - 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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int e^{n \tanh ^{-1}(a x)} (c-a c x) \, dx &=c \int (1-a x)^{1-\frac{n}{2}} (1+a x)^{n/2} \, dx\\ &=-\frac{2^{1+\frac{n}{2}} c (1-a x)^{2-\frac{n}{2}} \, _2F_1\left (2-\frac{n}{2},-\frac{n}{2};3-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (4-n)}\\ \end{align*}

Mathematica [A]  time = 0.0166896, size = 63, normalized size = 0.95 \[ \frac{c 2^{\frac{n}{2}+1} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (2-\frac{n}{2},-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (n-4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(1 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[2 - n/2, -n/2, 3 - n/2, (1 - a*x)/2])/(a*(-4 + n))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(Integral(a*x*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x)), x))

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

[Out]

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