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

Optimal. Leaf size=130 \[ \frac{c^2 2^{n/2} (1-a x)^{2-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},2-\frac{n}{2},3-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (4-n)}+\frac{4 c^2 (a x+1)^{n/2} (1-a x)^{-n/2} \text{Hypergeometric2F1}\left (2,\frac{n}{2},\frac{n+2}{2},\frac{a x+1}{1-a x}\right )}{a n} \]

[Out]

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

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Rubi [C]  time = 0.12373, antiderivative size = 71, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6131, 6129, 136} \[ \frac{c^2 2^{3-\frac{n}{2}} (a x+1)^{\frac{n+2}{2}} F_1\left (\frac{n+2}{2};\frac{n-4}{2},2;\frac{n+4}{2};\frac{1}{2} (a x+1),a x+1\right )}{a (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(3 - n/2)*c^2*(1 + a*x)^((2 + n)/2)*AppellF1[(2 + n)/2, (-4 + n)/2, 2, (4 + n)/2, (1 + a*x)/2, 1 + a*x])/(a
*(2 + n))

Rule 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, 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 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 e^{n \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{n \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \frac{(1-a x)^{2-\frac{n}{2}} (1+a x)^{n/2}}{x^2} \, dx}{a^2}\\ &=\frac{2^{3-\frac{n}{2}} c^2 (1+a x)^{\frac{2+n}{2}} F_1\left (\frac{2+n}{2};\frac{1}{2} (-4+n),2;\frac{4+n}{2};\frac{1}{2} (1+a x),1+a x\right )}{a (2+n)}\\ \end{align*}

Mathematica [B]  time = 0.479392, size = 262, normalized size = 2.02 \[ -\frac{c^2 e^{n \tanh ^{-1}(a x)} \left (a n^2 x \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \tanh ^{-1}(a x)}\right )-2 a n x e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+a (n-2) n x e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \tanh ^{-1}(a x)}\right )+2 a n x \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \tanh ^{-1}(a x)}\right )-4 a n x e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+4 a x \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \tanh ^{-1}(a x)}\right )-4 a x \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \tanh ^{-1}(a x)}\right )+n^2+2 n\right )}{a^2 n (n+2) x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((c^2*E^(n*ArcTanh[a*x])*(2*n + n^2 - 2*a*E^(2*ArcTanh[a*x])*n*x*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2
*ArcTanh[a*x])] + a*E^(2*ArcTanh[a*x])*(-2 + n)*n*x*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcTanh[a*x])]
+ 4*a*x*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcTanh[a*x])] + 2*a*n*x*Hypergeometric2F1[1, n/2, 1 + n/2,
-E^(2*ArcTanh[a*x])] - 4*a*x*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcTanh[a*x])] + a*n^2*x*Hypergeometric2F
1[1, n/2, 1 + n/2, E^(2*ArcTanh[a*x])] - 4*a*E^(2*ArcTanh[a*x])*n*x*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^
(2*ArcTanh[a*x])]))/(a^2*n*(2 + n)*x))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{ax}} \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((c - c/(a*x))^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 (\frac{{\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} x^{2}}, 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)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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