3.1332 \(\int \frac{e^{n \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx\)

Optimal. Leaf size=268 \[ -\frac{2 a n \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{3-n}{2},\frac{1-a x}{a x+1}\right )}{(1-n) \sqrt{1-a^2 x^2}}+\frac{a 2^{\frac{n+1}{2}} \sqrt{c-a^2 c x^2} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{1-n}{2},\frac{3-n}{2},\frac{1}{2} (1-a x)\right )}{(1-n) \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1-n}{2}}}{x \sqrt{1-a^2 x^2}} \]

[Out]

-(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2])) - (2*a*n*(1 - a*x)^
((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (1 - a*x)/(1
+ a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*a*(1 - a*x)^((1 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeomet
ric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/((1 - n)*Sqrt[1 - a^2*x^2])

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Rubi [C]  time = 0.233537, antiderivative size = 97, normalized size of antiderivative = 0.36, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 136} \[ \frac{a 2^{\frac{3}{2}-\frac{n}{2}} \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+3}{2}} F_1\left (\frac{n+3}{2};\frac{n-1}{2},2;\frac{n+5}{2};\frac{1}{2} (a x+1),a x+1\right )}{(n+3) \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(3/2 - n/2)*a*(1 + a*x)^((3 + n)/2)*Sqrt[c - a^2*c*x^2]*AppellF1[(3 + n)/2, (-1 + n)/2, 2, (5 + n)/2, (1 +
a*x)/2, 1 + a*x])/((3 + n)*Sqrt[1 - a^2*x^2])

Rule 6153

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

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 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \frac{e^{n \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{(1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{\frac{1}{2}+\frac{n}{2}}}{x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{2^{\frac{3}{2}-\frac{n}{2}} a (1+a x)^{\frac{3+n}{2}} \sqrt{c-a^2 c x^2} F_1\left (\frac{3+n}{2};\frac{1}{2} (-1+n),2;\frac{5+n}{2};\frac{1}{2} (1+a x),1+a x\right )}{(3+n) \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.423031, size = 138, normalized size = 0.51 \[ -\frac{c \sqrt{1-a^2 x^2} e^{n \tanh ^{-1}(a x)} \left (2 a x e^{\tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \tanh ^{-1}(a x)}\right )+2 a n x e^{\tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \tanh ^{-1}(a x)}\right )+(n+1) \sqrt{1-a^2 x^2}\right )}{(n+1) x \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{2}}\sqrt{-{a}^{2}c{x}^{2}+c}}\, 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)^(1/2)/x^2,x)

[Out]

int(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x^2,x)

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

[Out]

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

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} e^{n \operatorname{atanh}{\left (a x \right )}}}{x^{2}}\, dx \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)**(1/2)/x**2,x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x))/x**2, x)

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

[Out]

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