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

Optimal. Leaf size=56 \[ \frac{2 x \sqrt{1-a x} F_1\left (\frac{3}{2};\frac{n+1}{2},-\frac{n}{2};\frac{5}{2};a x,-a x\right )}{3 \sqrt{c-\frac{c}{a x}}} \]

[Out]

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

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Rubi [A]  time = 0.154303, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6134, 6129, 133} \[ \frac{2 x \sqrt{1-a x} F_1\left (\frac{3}{2};\frac{n+1}{2},-\frac{n}{2};\frac{5}{2};a x,-a x\right )}{3 \sqrt{c-\frac{c}{a x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6134

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

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 133

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

Rubi steps

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

Mathematica [F]  time = 180.006, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [F]  time = 0.121, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{\frac{1}{\sqrt{c-{\frac{c}{ax}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/(c-c/a/x)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^(1/2),x, algorithm="giac")

[Out]

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