### 3.928 $$\int e^{n \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2}) \, dx$$

Optimal. Leaf size=154 $\frac{4 c \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n-2}{2}} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (2-n)}-\frac{c 2^{\frac{n}{2}+1} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{a (2-n)}$

[Out]

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

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Rubi [C]  time = 0.0652096, antiderivative size = 81, normalized size of antiderivative = 0.53, 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 = {6194, 136} $-\frac{c 2^{2-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n+4}{2}} F_1\left (\frac{n+4}{2};\frac{n-2}{2},2;\frac{n+6}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (n+4)}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((2^(2 - n/2)*c*(1 + 1/(a*x))^((4 + n)/2)*AppellF1[(4 + n)/2, (-2 + n)/2, 2, (6 + n)/2, (a + x^(-1))/(2*a), 1
+ 1/(a*x)])/(a*(4 + n)))

Rule 6194

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

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^2 x^2}\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 )^{1+\frac{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{4+n}{2}} F_1\left (\frac{4+n}{2};\frac{1}{2} (-2+n),2;\frac{6+n}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right )}{a (4+n)}\\ \end{align*}

Mathematica [A]  time = 0.239939, size = 123, normalized size = 0.8 $\frac{c e^{n \coth ^{-1}(a x)} \left (n e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \coth ^{-1}(a x)}\right )+(n+2) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \coth ^{-1}(a x)}\right )+4 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \coth ^{-1}(a x)}\right )+a n x+2 a x\right )}{a (n+2)}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*E^(n*ArcCoth[a*x])*(2*a*x + a*n*x + E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCot
h[a*x])] + (2 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])] + 4*E^(2*ArcCoth[a*x])*Hypergeometri
c2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]))/(a*(2 + n))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) \, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((c - c/(a^2*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 x^{2} - c\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*arccoth(a*x))*(c-c/a^2/x^2),x, algorithm="fricas")

[Out]

integral((a^2*c*x^2 - c)*((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 \left (\int a^{2} e^{n \operatorname{acoth}{\left (a x \right )}}\, dx + \int - \frac{e^{n \operatorname{acoth}{\left (a x \right )}}}{x^{2}}\, dx\right )}{a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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