\(\int e^{\text {csch}^{-1}(a x)} x^m \, dx\) [22]

Optimal result

Integrand size = 10, antiderivative size = 52 \[ \int e^{\text {csch}^{-1}(a x)} x^m \, dx=\frac {x^m}{a m}+\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},-\frac {1}{a^2 x^2}\right )}{1+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04 \[ \int e^{\text {csch}^{-1}(a x)} x^m \, dx=\frac {x^m}{a m}+\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1-m),1+\frac {1}{2} (-1-m),-\frac {1}{a^2 x^2}\right )}{1+m} \] Input:

Integrate[E^ArcCsch[a*x]*x^m,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6890, 15, 862, 278}

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.

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

\(\Big \downarrow \) 6890

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 862

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

\(\Big \downarrow \) 278

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

Input:

Int[E^ArcCsch[a*x]*x^m,x]
 

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 862

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ 
(-1))*(c*x)^(m + 1)*(1/x)^(m + 1)   Subst[Int[(a + b/x^n)^p/x^(m + 2), x], 
x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]
 

rule 6890

Int[E^ArcCsch[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[1/a   Int[x^( 
m - p), x], x] + Int[x^m*Sqrt[1 + 1/(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, 
x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 2.94 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.25 \[ \int e^{\text {csch}^{-1}(a x)} x^m \, dx=- \frac {a^{m} a^{- m - 1} x^{m} \Gamma \left (- \frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} \\ \frac {m}{2} + 1 \end {matrix}\middle | {a^{2} x^{2} e^{i \pi }} \right )}}{2 \Gamma \left (1 - \frac {m}{2}\right )} - \frac {\begin {cases} - \log {\left (x \right )} & \text {for}\: m = 0 \\- \frac {x^{m}}{m} & \text {otherwise} \end {cases}}{a} \] Input:

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

Output:

-a**m*a**(-m - 1)*x**m*gamma(-m/2)*hyper((-1/2, m/2), (m/2 + 1,), a**2*x** 
2*exp_polar(I*pi))/(2*gamma(1 - m/2)) - Piecewise((-log(x), Eq(m, 0)), (-x 
**m/m, True))/a
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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