3.1308 \(\int e^{n \tanh ^{-1}(a x)} (c-a^2 c x^2) \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0441174, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6140, 69} \[ -\frac{c 2^{\frac{n}{2}+2} (1-a x)^{2-\frac{n}{2}} \, _2F_1\left (-\frac{n}{2}-1,2-\frac{n}{2};3-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (4-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 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)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x)^{1-\frac{n}{2}} (1+a x)^{1+\frac{n}{2}} \, dx\\ &=-\frac{2^{2+\frac{n}{2}} c (1-a x)^{2-\frac{n}{2}} \, _2F_1\left (-1-\frac{n}{2},2-\frac{n}{2};3-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (4-n)}\\ \end{align*}

Mathematica [A]  time = 0.0190884, size = 65, normalized size = 0.96 \[ \frac{c 2^{\frac{n}{2}+2} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,2-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (n-4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \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))*(-a^2*c*x^2+c),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{2} c x^{2} - 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^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - c \left (\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**2*c*x**2+c),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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