3.1309 \(\int e^{n \tanh ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=70 \[ -\frac{c^2 2^{\frac{n}{2}+3} (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-2,3-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (6-n)} \]

[Out]

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

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Rubi [A]  time = 0.0536608, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 69} \[ -\frac{c^2 2^{\frac{n}{2}+3} (1-a x)^{3-\frac{n}{2}} \, _2F_1\left (-\frac{n}{2}-2,3-\frac{n}{2};4-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a (6-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0205565, size = 67, normalized size = 0.96 \[ \frac{c^2 2^{\frac{n}{2}+3} (1-a x)^{3-\frac{n}{2}} \text{Hypergeometric2F1}\left (-\frac{n}{2}-2,3-\frac{n}{2},4-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (n-6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.144, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{2}\, 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)^2,x)

[Out]

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

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

[Out]

integrate((a^2*c*x^2 - 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^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + 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^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + 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^{2} x^{2} e^{n \operatorname{atanh}{\left (a x \right )}}\, dx + \int a^{4} x^{4} 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)**2,x)

[Out]

c**2*(Integral(-2*a**2*x**2*exp(n*atanh(a*x)), x) + Integral(a**4*x**4*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^{2} c x^{2} - 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^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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