3.542 \(\int e^{n \coth ^{-1}(a x)} (c-\frac{c}{a x}) \, dx\)

Optimal. Leaf size=185 \[ -\frac{c 2^{n/2} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},1-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{a (2-n)}-\frac{2 c (1-n) \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{-n/2} \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n+2}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{a n}+c x \left (\frac{1}{a x}+1\right )^{n/2} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \]

[Out]

c*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^(n/2)*x - (2*c*(1 - n)*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (
2 + n)/2, (a + x^(-1))/(a - x^(-1))])/(a*n*(1 - 1/(a*x))^(n/2)) - (2^(n/2)*c*(1 - 1/(a*x))^(1 - n/2)*Hypergeom
etric2F1[1 - n/2, 1 - n/2, 2 - n/2, (a - x^(-1))/(2*a)])/(a*(2 - n))

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Rubi [C]  time = 0.0645511, antiderivative size = 81, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6179, 136} \[ -\frac{c 2^{2-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} F_1\left (\frac{n+2}{2};\frac{n-2}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 6179

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1 + (d*x)/c)^
p*(1 + x/a)^(n/2))/(x^2*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0
] &&  !IntegerQ[n/2] && (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 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{2^{2-\frac{n}{2}} c \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} F_1\left (\frac{2+n}{2};\frac{1}{2} (-2+n),2;\frac{4+n}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.350406, size = 155, normalized size = 0.84 \[ \frac{c e^{n \coth ^{-1}(a x)} \left (n \left (-e^{2 \coth ^{-1}(a x)}\right ) \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \coth ^{-1}(a x)}\right )+(n-1) n e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+(n+2) \left (\text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \coth ^{-1}(a x)}\right )+(n-1) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+a n x\right )\right )}{a n (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*E^(n*ArcCoth[a*x])*(-(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]) + E
^(2*ArcCoth[a*x])*(-1 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*n*x + Hyp
ergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])] + (-1 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCot
h[a*x])])))/(a*n*(2 + n))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{ax}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))*(c-c/a/x),x)

[Out]

int(exp(n*arccoth(a*x))*(c-c/a/x),x)

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

[Out]

integrate((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 (\frac{{\left (a c x - c\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*(c-c/a/x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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