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

Optimal. Leaf size=54 \[ \frac{2 x \sqrt{c-\frac{c}{a x}} F_1\left (\frac{1}{2};\frac{n-1}{2},-\frac{n}{2};\frac{3}{2};a x,-a x\right )}{\sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.150397, antiderivative size = 54, 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} \[ \frac{2 x \sqrt{c-\frac{c}{a x}} F_1\left (\frac{1}{2};\frac{n-1}{2},-\frac{n}{2};\frac{3}{2};a x,-a x\right )}{\sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c - c/(a*x)]*x*AppellF1[1/2, (-1 + n)/2, -n/2, 3/2, a*x, -(a*x)])/Sqrt[1 - 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 e^{n \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{e^{n \tanh ^{-1}(a x)} \sqrt{1-a x}}{\sqrt{x}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{(1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{n/2}}{\sqrt{x}} \, dx}{\sqrt{1-a x}}\\ &=\frac{2 \sqrt{c-\frac{c}{a x}} x F_1\left (\frac{1}{2};\frac{1}{2} (-1+n),-\frac{n}{2};\frac{3}{2};a x,-a x\right )}{\sqrt{1-a x}}\\ \end{align*}

Mathematica [F]  time = 180.007, 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 = 0.122, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}\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 \sqrt{c - \frac{c}{a x}} \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))*(c-c/a/x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c - c/(a*x))*((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 (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n} \sqrt{\frac{a c x - c}{a x}}, 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 + 1)/(a*x - 1))^(1/2*n)*sqrt((a*c*x - c)/(a*x)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

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

[Out]

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