\(\int e^{2 \text {csch}^{-1}(a x)} x^m \, dx\) [21]

Optimal result

Integrand size = 12, antiderivative size = 64 \[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=-\frac {2 x^{-1+m}}{a^2 (1-m)}+\frac {x^{1+m}}{1+m}+\frac {2 x^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},-\frac {1}{a^2 x^2}\right )}{a m} \] Output:

-2*x^(-1+m)/a^2/(1-m)+x^(1+m)/(1+m)+2*x^m*hypergeom([-1/2, -1/2*m],[1-1/2* 
m],-1/a^2/x^2)/a/m
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=x^m \left (\frac {2}{a^2 (-1+m) x}+\frac {x}{1+m}+\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},-\frac {1}{a^2 x^2}\right )}{a m}\right ) \] Input:

Integrate[E^(2*ArcCsch[a*x])*x^m,x]
 

Output:

x^m*(2/(a^2*(-1 + m)*x) + x/(1 + m) + (2*Hypergeometric2F1[-1/2, -1/2*m, 1 
 - m/2, -(1/(a^2*x^2))])/(a*m))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6892, 7293, 2009}

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.

\(\displaystyle \int x^m e^{2 \text {csch}^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6892

\(\displaystyle \int \left (\sqrt {\frac {1}{a^2 x^2}+1}+\frac {1}{a x}\right )^2 x^mdx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^{m-2}}{a^2}+\frac {2 \sqrt {\frac {1}{a^2 x^2}+1} x^{m-1}}{a}+x^m\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 x^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},-\frac {1}{a^2 x^2}\right )}{a m}-\frac {2 x^{m-1}}{a^2 (1-m)}+\frac {x^{m+1}}{m+1}\)

Input:

Int[E^(2*ArcCsch[a*x])*x^m,x]
 

Output:

(-2*x^(-1 + m))/(a^2*(1 - m)) + x^(1 + m)/(1 + m) + (2*x^m*Hypergeometric2 
F1[-1/2, -1/2*m, 1 - m/2, -(1/(a^2*x^2))])/(a*m)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6892

Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 
1/u^2])^n, x] /; FreeQ[m, x] && IntegerQ[n]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int {\left (\frac {1}{a x}+\sqrt {1+\frac {1}{a^{2} x^{2}}}\right )}^{2} x^{m}d x\]

Input:

int((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^m,x)
 

Output:

int((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^m,x)
 

Fricas [F]

\[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=\int { x^{m} {\left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + \frac {1}{a x}\right )}^{2} \,d x } \] Input:

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^m,x, algorithm="fricas")
 

Output:

integral((2*a*x*x^m*sqrt((a^2*x^2 + 1)/(a^2*x^2)) + (a^2*x^2 + 2)*x^m)/(a^ 
2*x^2), x)
 

Sympy [A] (verification not implemented)

Time = 3.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.19 \[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=\begin {cases} \frac {x^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases} - \frac {x^{m} \Gamma \left (- \frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {m}{2} \\ 1 - \frac {m}{2} \end {matrix}\middle | {\frac {e^{i \pi }}{a^{2} x^{2}}} \right )}}{a \Gamma \left (1 - \frac {m}{2}\right )} + \frac {2 \left (\begin {cases} \frac {x^{m}}{m x - x} & \text {for}\: m \neq 1 \\\frac {x^{m} \log {\left (x \right )}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} \] Input:

integrate((1/a/x+(1+1/a**2/x**2)**(1/2))**2*x**m,x)
 

Output:

Piecewise((x**(m + 1)/(m + 1), Ne(m, -1)), (log(x), True)) - x**m*gamma(-m 
/2)*hyper((-1/2, -m/2), (1 - m/2,), exp_polar(I*pi)/(a**2*x**2))/(a*gamma( 
1 - m/2)) + 2*Piecewise((x**m/(m*x - x), Ne(m, 1)), (x**m*log(x)/x, True)) 
/a**2
 

Maxima [F(-2)]

Exception generated. \[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=\text {Exception raised: ValueError} \] Input:

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^m,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(m-2>0)', see `assume?` for more 
details)Is
 

Giac [F(-2)]

Exception generated. \[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=\text {Exception raised: TypeError} \] Input:

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^m,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=\int x^m\,{\left (\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {1}{a\,x}\right )}^2 \,d x \] Input:

int(x^m*((1/(a^2*x^2) + 1)^(1/2) + 1/(a*x))^2,x)
 

Output:

int(x^m*((1/(a^2*x^2) + 1)^(1/2) + 1/(a*x))^2, x)
 

Reduce [F]

\[ \int e^{2 \text {csch}^{-1}(a x)} x^m \, dx=\frac {x^{m} a^{2} m \,x^{2}-x^{m} a^{2} x^{2}+2 x^{m} m +2 x^{m}+2 \left (\int \frac {x^{m} \sqrt {a^{2} x^{2}+1}}{x^{2}}d x \right ) m^{2} x -2 \left (\int \frac {x^{m} \sqrt {a^{2} x^{2}+1}}{x^{2}}d x \right ) x}{a^{2} x \left (m^{2}-1\right )} \] Input:

int((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^m,x)
 

Output:

(x**m*a**2*m*x**2 - x**m*a**2*x**2 + 2*x**m*m + 2*x**m + 2*int((x**m*sqrt( 
a**2*x**2 + 1))/x**2,x)*m**2*x - 2*int((x**m*sqrt(a**2*x**2 + 1))/x**2,x)* 
x)/(a**2*x*(m**2 - 1))