### 3.755 $$\int \frac{e^{n \coth ^{-1}(a x)}}{x (c-a^2 c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=277 $-\frac{a^3 2^{\frac{n+1}{2}} x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (\frac{1-n}{2},\frac{1-n}{2},\frac{3-n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}}$

[Out]

-((a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^3)/((1 + n)*(c - a^2*c*
x^2)^(3/2))) + (a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^3)/((1 - n^
2)*(c - a^2*c*x^2)^(3/2)) - (2^((1 + n)/2)*a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*x^3*Hypergeom
etric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (a - x^(-1))/(2*a)])/((1 - n)*(c - a^2*c*x^2)^(3/2))

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Rubi [A]  time = 0.373535, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.185, Rules used = {6192, 6195, 89, 79, 69} $-\frac{a^3 2^{\frac{n+1}{2}} x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \, _2F_1\left (\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{a^3 x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) \left (c-a^2 c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(n*ArcCoth[a*x])/(x*(c - a^2*c*x^2)^(3/2)),x]

[Out]

-((a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^3)/((1 + n)*(c - a^2*c*
x^2)^(3/2))) + (a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^3)/((1 - n^
2)*(c - a^2*c*x^2)^(3/2)) - (2^((1 + n)/2)*a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*x^3*Hypergeom
etric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (a - x^(-1))/(2*a)])/((1 - n)*(c - a^2*c*x^2)^(3/2))

Rule 6192

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

Rule 6195

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^4} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int x^2 \left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \left (\frac{n}{a}+\frac{(1+n) x}{a^2}\right ) \, dx,x,\frac{1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{\frac{1}{2} (-1+n)} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{\left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{2^{\frac{1+n}{2}} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} x^3 \, _2F_1\left (\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{(1-n) \left (c-a^2 c x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.824422, size = 127, normalized size = 0.46 $\frac{e^{n \coth ^{-1}(a x)} \left (a x \sqrt{1-\frac{1}{a^2 x^2}} (a n x-1)-2 (n-1) \left (a^2 x^2-1\right ) e^{\coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )}{a c (n-1) (n+1) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcCoth[a*x])/(x*(c - a^2*c*x^2)^(3/2)),x]

[Out]

(E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*n*x) - 2*E^ArcCoth[a*x]*(-1 + n)*(-1 + a^2*x^2)*Hyperge
ometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])]))/(a*c*(-1 + n)*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[
c - a^2*c*x^2])

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Maple [F]  time = 0.309, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{x} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(3/2),x)

[Out]

int(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(3/2),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}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/((-a^2*c*x^2 + c)^(3/2)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{5} - 2 \, a^{2} c^{2} x^{3} + c^{2} x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a^4*c^2*x^5 - 2*a^2*c^2*x^3 + c^2*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*acoth(a*x))/x/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Timed out

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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}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))/x/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(1/2*n)/((-a^2*c*x^2 + c)^(3/2)*x), x)