\(\int e^{\text {sech}^{-1}(\frac {a}{x})} x^m \, dx\) [63]

Optimal result

Integrand size = 12, antiderivative size = 74 \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=\frac {x^{2+m}}{a (2+m)}+\frac {x^{1+m} \sqrt {-1+\frac {x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {x^2}{a^2}\right )}{(1+m) \sqrt {1-\frac {x}{a}}} \] Output:

x^(2+m)/a/(2+m)+x^(1+m)*(-1+x/a)^(1/2)*hypergeom([-1/2, 1/2+1/2*m],[3/2+1/ 
2*m],x^2/a^2)/(1+m)/(1-x/a)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.76 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.88 \[ \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx=-\frac {2^{-1-m} a e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} \left (\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{1+e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}}\right )^{-1-m} \left (\frac {a}{x}\right )^m x^m \left (-\left ((-2+m) \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},1-\frac {m}{2},-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )\right )+e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )} m \operatorname {Hypergeometric2F1}\left (1,2+\frac {m}{2},2-\frac {m}{2},-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )\right )}{(-2+m) m} \] Input:

Integrate[E^ArcSech[a/x]*x^m,x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6889, 15, 791, 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.

\(\displaystyle \int x^m e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} \, dx\)

\(\Big \downarrow \) 6889

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 791

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

\(\Big \downarrow \) 862

\(\displaystyle \frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} \left (\frac {1}{x}\right )^m x^m \int \frac {\left (\frac {1}{x}\right )^{-m-3}}{\sqrt {1-\frac {a^2}{x^2}}}d\frac {1}{x}}{a (m+1)}-\frac {x^{m+2}}{a (m+1) (m+2)}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1}\)

\(\Big \downarrow \) 278

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

Input:

Int[E^ArcSech[a/x]*x^m,x]
 

Output:

(E^ArcSech[a/x]*x^(1 + m))/(1 + m) - x^(2 + m)/(a*(1 + m)*(2 + m)) - (Sqrt 
[(1 + a/x)^(-1)]*Sqrt[1 + a/x]*x^(2 + m)*Hypergeometric2F1[1/2, (-2 - m)/2 
, -1/2*m, a^2/x^2])/(a*(1 + m)*(2 + 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 791

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_) 
^(n_))^(p_), x_Symbol] :> Int[(c*x)^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; Free 
Q[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] 
 || (GtQ[a1, 0] && GtQ[a2, 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 6889

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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