3.1331 \(\int \frac{e^{n \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx\)

Optimal. Leaf size=269 \[ \frac{2^{\frac{n+1}{2}} n \sqrt{c-a^2 c x^2} (1-a x)^{\frac{3-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{3-n}{2},\frac{5-n}{2},\frac{1}{2} (1-a x)\right )}{\left (n^2-4 n+3\right ) \sqrt{1-a^2 x^2}}+\frac{2 \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a x+1}{1-a x}\right )}{(1-n) \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{3-n}{2}}}{(1-n) \sqrt{1-a^2 x^2}} \]

[Out]

-(((1 - a*x)^((3 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[c - a^2*c*x^2])/((1 - n)*Sqrt[1 - a^2*x^2])) + (2*(1 - a*
x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, (1 + a*x
)/(1 - a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*n*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hyperge
ometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/((3 - 4*n + n^2)*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.293015, antiderivative size = 299, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6153, 6150, 105, 69, 131} \[ \frac{2 \sqrt{c-a^2 c x^2} (a x+1)^{\frac{n+1}{2}} (1-a x)^{\frac{1}{2} (-n-1)} \, _2F_1\left (1,\frac{1}{2} (-n-1);\frac{1-n}{2};\frac{1-a x}{a x+1}\right )}{(n+1) \sqrt{1-a^2 x^2}}-\frac{2^{\frac{n+3}{2}} \sqrt{c-a^2 c x^2} (1-a x)^{\frac{1}{2} (-n-1)} \, _2F_1\left (\frac{1}{2} (-n-1),\frac{1}{2} (-n-1);\frac{1-n}{2};\frac{1}{2} (1-a x)\right )}{(n+1) \sqrt{1-a^2 x^2}}+\frac{2^{\frac{n+3}{2}} \sqrt{c-a^2 c x^2} (1-a x)^{\frac{1-n}{2}} \, _2F_1\left (\frac{1}{2} (-n-1),\frac{1-n}{2};\frac{3-n}{2};\frac{1}{2} (1-a x)\right )}{(1-n) \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (-1 - n)/2, (1 - n)/2
, (1 - a*x)/(1 + a*x)])/((1 + n)*Sqrt[1 - a^2*x^2]) - (2^((3 + n)/2)*(1 - a*x)^((-1 - n)/2)*Sqrt[c - a^2*c*x^2
]*Hypergeometric2F1[(-1 - n)/2, (-1 - n)/2, (1 - n)/2, (1 - a*x)/2])/((1 + n)*Sqrt[1 - a^2*x^2]) + (2^((3 + n)
/2)*(1 - a*x)^((1 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2]
)/((1 - 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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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]))

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

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

Mathematica [A]  time = 0.156866, size = 207, normalized size = 0.77 \[ \frac{2 \sqrt{c-a^2 c x^2} (1-a x)^{\frac{1}{2} (-n-1)} \left ((n-1) (a x+1)^{\frac{n+1}{2}} \text{Hypergeometric2F1}\left (1,-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )+2^{\frac{n+1}{2}} \left ((n+1) (a x-1) \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{n}{2},\frac{3}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )-(n-1) \text{Hypergeometric2F1}\left (-\frac{n}{2}-\frac{1}{2},-\frac{n}{2}-\frac{1}{2},\frac{1}{2}-\frac{n}{2},\frac{1}{2}-\frac{a x}{2}\right )\right )\right )}{\left (n^2-1\right ) \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{x}\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,x)

[Out]

int(exp(n*arctanh(a*x))*(-a^2*c*x^2+c)^(1/2)/x,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}\,{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,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/x, 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}, 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,x, algorithm="fricas")

[Out]

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

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x))/x, 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}\,{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,x, algorithm="giac")

[Out]

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