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

Optimal. Leaf size=68 \[ -\frac{c^2 2^{\frac{n}{2}+1} (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (3-\frac{n}{2},-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (6-n)} \]

[Out]

-((2^(1 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[3 - n/2, -n/2, 4 - n/2, (1 - a*x)/2])/(a*(6 - n)))

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Rubi [A]  time = 0.0489045, antiderivative size = 68, 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, 69} \[ -\frac{c^2 2^{\frac{n}{2}+1} (1-a x)^{3-\frac{n}{2}} \, _2F_1\left (3-\frac{n}{2},-\frac{n}{2};4-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (6-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(1 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[3 - n/2, -n/2, 4 - n/2, (1 - a*x)/2])/(a*(6 - 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)^2 \, dx &=c^2 \int (1-a x)^{2-\frac{n}{2}} (1+a x)^{n/2} \, dx\\ &=-\frac{2^{1+\frac{n}{2}} c^2 (1-a x)^{3-\frac{n}{2}} \, _2F_1\left (3-\frac{n}{2},-\frac{n}{2};4-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (6-n)}\\ \end{align*}

Mathematica [A]  time = 0.0172727, size = 65, normalized size = 0.96 \[ \frac{c^2 2^{\frac{n}{2}+1} (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (3-\frac{n}{2},-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (n-6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(1 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[3 - n/2, -n/2, 4 - n/2, (1 - a*x)/2])/(a*(-6 + n))

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Maple [F]  time = 0.159, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{2}\, dx \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]

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

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

[Out]

integrate((a*c*x - c)^2*((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^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\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)^2,x, algorithm="fricas")

[Out]

integral((a^2*c^2*x^2 - 2*a*c^2*x + c^2)*((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^{2} \left (\int - 2 a x e^{n \operatorname{atanh}{\left (a x \right )}}\, dx + \int a^{2} x^{2} 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)**2,x)

[Out]

c**2*(Integral(-2*a*x*exp(n*atanh(a*x)), x) + Integral(a**2*x**2*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 )}^{2} \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)^2,x, algorithm="giac")

[Out]

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