### 3.751 $$\int \frac{e^{n \coth ^{-1}(a x)} x^3}{(c-a^2 c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=359 $-\frac{2 n x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (n^2+2 n+2\right ) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac{x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{(n+2) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.332594, antiderivative size = 363, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6192, 6194, 129, 155, 12, 131} $\frac{2 n x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (3-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (n^2+2 n+2\right ) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac{x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{(n+2) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((2 + n)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^3)/(a*(1 + n)*(c -
a^2*c*x^2)^(3/2))) + ((2 + 2*n + n^2)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n
)/2)*x^3)/(a*(1 - n)*(1 + n)*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 +
1/(a*x))^((-1 + n)/2)*x^4)/(c - a^2*c*x^2)^(3/2) + (2*n*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((3 - n)/2)*(1
+ 1/(a*x))^((-3 + n)/2)*x^3*Hypergeometric2F1[1, (3 - n)/2, (5 - n)/2, (a - x^(-1))/(a + x^(-1))])/(a*(3 - 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 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 129

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

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^3}{\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}} \, 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 \frac{\left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\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^4}{\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 \frac{\left (-\frac{n}{a}-\frac{2 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\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^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (a \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{\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 (1+n)}{a^2}+\frac{(2+n) x}{a^3}\right )}{x} \, dx,x,\frac{1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2+2 n+n^2\right ) \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}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\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^4}{\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 \frac{n \left (1-n^2\right ) \left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{a^3 x} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2+2 n+n^2\right ) \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}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\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^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (n \left (1-n^2\right ) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \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}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2+2 n+n^2\right ) \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}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\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^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{2 n \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^3 \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (3-n) \left (c-a^2 c x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.552972, size = 133, normalized size = 0.37 $\frac{\frac{c (a n x-1) e^{n \coth ^{-1}(a x)}}{n^2-1}-\frac{c \left (a^2 x^2-1\right ) \left (\frac{2 n e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )}{a x \sqrt{1-\frac{1}{a^2 x^2}}}+(n+1) e^{n \coth ^{-1}(a x)}\right )}{n+1}}{a^4 c^2 \sqrt{c-a^2 c x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.307, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{x}^{3} \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^3/(-a^2*c*x^2+c)^(3/2),x)

[Out]

int(exp(n*arccoth(a*x))*x^3/(-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{x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(3/2), 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} x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*c*x^2 + c)*x^3*((a*x - 1)/(a*x + 1))^(1/2*n)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), 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**3/(-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{x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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