\(\int e^{\text {sech}^{-1}(a x)} x^m \, dx\) [62]

Optimal result

Integrand size = 10, antiderivative size = 78 \[ \int e^{\text {sech}^{-1}(a x)} x^m \, dx=\frac {x^m}{a m}+\frac {\sqrt {-1+\frac {1}{a x}} x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{(1+m) \sqrt {1-\frac {1}{a x}}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.86 \[ \int e^{\text {sech}^{-1}(a x)} x^m \, dx=-\frac {2^{1+m} e^{2 \text {sech}^{-1}(a x)} \left (\frac {e^{\text {sech}^{-1}(a x)}}{1+e^{2 \text {sech}^{-1}(a x)}}\right )^m \left (1+e^{2 \text {sech}^{-1}(a x)}\right )^m x^m (a x)^{-m} \left (-\left ((4+m) \operatorname {Hypergeometric2F1}\left (1+\frac {m}{2},2+m,2+\frac {m}{2},-e^{2 \text {sech}^{-1}(a x)}\right )\right )+e^{2 \text {sech}^{-1}(a x)} (2+m) \operatorname {Hypergeometric2F1}\left (2+\frac {m}{2},2+m,3+\frac {m}{2},-e^{2 \text {sech}^{-1}(a x)}\right )\right )}{a (2+m) (4+m)} \] Input:

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

Output:

-((2^(1 + m)*E^(2*ArcSech[a*x])*(E^ArcSech[a*x]/(1 + E^(2*ArcSech[a*x])))^ 
m*(1 + E^(2*ArcSech[a*x]))^m*x^m*(-((4 + m)*Hypergeometric2F1[1 + m/2, 2 + 
 m, 2 + m/2, -E^(2*ArcSech[a*x])]) + E^(2*ArcSech[a*x])*(2 + m)*Hypergeome 
tric2F1[2 + m/2, 2 + m, 3 + m/2, -E^(2*ArcSech[a*x])]))/(a*(2 + m)*(4 + m) 
*(a*x)^m))
 

Rubi [A] (warning: unable to verify)

Time = 0.44 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6889, 15, 135, 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.

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

\(\Big \downarrow \) 6889

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 135

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

x^m/(a*m*(1 + m)) + (E^ArcSech[a*x]*x^(1 + m))/(1 + m) + (x^m*Sqrt[(1 + a* 
x)^(-1)]*Sqrt[1 + a*x]*Hypergeometric2F1[1/2, m/2, (2 + m)/2, a^2*x^2])/(a 
*m*(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 135

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

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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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