\(\int \frac {e^{n \coth ^{-1}(a x)} x^2}{(c-a^2 c x^2)^{3/2}} \, dx\) [726]

Optimal result

Integrand size = 27, antiderivative size = 164 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}} \] Output:

-exp(n*arccoth(a*x))*(-a*x+n)/a^3/c/(-n^2+1)/(-a^2*c*x^2+c)^(1/2)-2*(1-1/a 
^2/x^2)^(1/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x*hypergeom([1, 
 1/2-1/2*n],[3/2-1/2*n],(a-1/x)/(a+1/x))/a^2/c/(1-n)/(-a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.77 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (-n+a x)+2 e^{\coth ^{-1}(a x)} (-1+n) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{a^4 c (-1+n) (1+n) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6743, 6746, 6749, 141}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6743

\(\displaystyle -\frac {\int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}}dx}{a^2 c}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 6746

\(\displaystyle -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}} x}dx}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 6749

\(\displaystyle \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \int \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (1+\frac {1}{a x}\right )^{\frac {n-1}{2}} xd\frac {1}{x}}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 141

\(\displaystyle -\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 c (1-n) \sqrt {c-a^2 c x^2}}-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 141

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

rule 6743

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(x_)^2*((c_) + (d_.)*(x_)^2)^(p_), x_Symb 
ol] :> Simp[(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcCoth[a*x])/(a^ 
3*c*(n^2 - 4*(p + 1)^2))), x] - Simp[(n^2 + 2*(p + 1))/(a^2*c*(n^2 - 4*(p + 
 1)^2))   Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c 
, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] && LeQ[p, -1] && NeQ[n^ 
2 + 2*(p + 1), 0] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !Integer 
Q[n])
 

rule 6746

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(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 6749

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x 
_Symbol] :> Simp[-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]
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{2}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integral(sqrt(-a^2*c*x^2 + c)*x^2*((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)
 

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(exp(n*acoth(a*x))*x**2/(-a**2*c*x**2+c)**(3/2),x)
 

Output:

Integral(x**2*exp(n*acoth(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\int \frac {e^{\mathit {acoth} \left (a x \right ) n} x^{2}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c} \] Input:

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

Output:

( - int((e**(acoth(a*x)*n)*x**2)/(sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( 
 - a**2*x**2 + 1)),x))/(sqrt(c)*c)