3.1.60 \(\int e^{\text {sech}^{-1}(a x^p)} x^m \, dx\) [60]

3.1.60.1 Optimal result

Integrand size = 12, antiderivative size = 133 \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {e^{\text {sech}^{-1}\left (a x^p\right )} x^{1+m}}{1+m}+\frac {p x^{1+m-p}}{a (1+m) (1+m-p)}+\frac {p x^{1+m-p} \sqrt {\frac {1}{1+a x^p}} \sqrt {1+a x^p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m-p}{2 p},\frac {1+m+p}{2 p},a^2 x^{2 p}\right )}{a (1+m) (1+m-p)} \]

output

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

3.1.60.2 Mathematica [A] (warning: unable to verify)

Time = 4.33 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.40 \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\frac {2^{\frac {1+m}{p}} e^{\text {sech}^{-1}\left (a x^p\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^p\right )}}\right )^{\frac {1+m}{p}} x^{1+m} \left (a x^p\right )^{-\frac {1+m}{p}} \left (-e^{2 \text {sech}^{-1}\left (a x^p\right )} (1+m+p) \operatorname {Hypergeometric2F1}\left (1,-\frac {1+m-3 p}{2 p},\frac {1+m+5 p}{2 p},-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )+(1+m+3 p) \operatorname {Hypergeometric2F1}\left (1,1-\frac {1+m+p}{2 p},\frac {1+m+3 p}{2 p},-e^{2 \text {sech}^{-1}\left (a x^p\right )}\right )\right )}{(1+m+p) (1+m+3 p)} \]

input

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

output

(2^((1 + m)/p)*E^ArcSech[a*x^p]*(E^ArcSech[a*x^p]/(1 + E^(2*ArcSech[a*x^p] 
)))^((1 + m)/p)*x^(1 + m)*(-(E^(2*ArcSech[a*x^p])*(1 + m + p)*Hypergeometr 
ic2F1[1, -1/2*(1 + m - 3*p)/p, (1 + m + 5*p)/(2*p), -E^(2*ArcSech[a*x^p])] 
) + (1 + m + 3*p)*Hypergeometric2F1[1, 1 - (1 + m + p)/(2*p), (1 + m + 3*p 
)/(2*p), -E^(2*ArcSech[a*x^p])]))/((1 + m + p)*(1 + m + 3*p)*(a*x^p)^((1 + 
 m)/p))
 

3.1.60.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6889, 15, 791, 888}

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.

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

\(\Big \downarrow \) 6889

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 791

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

\(\Big \downarrow \) 888

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

input

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

output

(E^ArcSech[a*x^p]*x^(1 + m))/(1 + m) + (p*x^(1 + m - p))/(a*(1 + m)*(1 + m 
 - p)) + (p*x^(1 + m - p)*Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*Hypergeom 
etric2F1[1/2, (1 + m - p)/(2*p), (1 + m + p)/(2*p), a^2*x^(2*p)])/(a*(1 + 
m)*(1 + m - p))
 

3.1.60.3.1 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 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 888

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

3.1.60.4 Maple [F]

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

input

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

output

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

3.1.60.5 Fricas [F(-2)]

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

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

3.1.60.6 Sympy [F]

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

input

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

output

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

3.1.60.7 Maxima [F(-2)]

Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^p\right )} x^m \, dx=\text {Exception raised: ValueError} \]

input

integrate((1/a/(x^p)+(1/a/(x^p)-1)^(1/2)*(1/a/(x^p)+1)^(1/2))*x^m,x, algor 
ithm="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-p>0)', see `assume?` for more 
details)Is
 

3.1.60.8 Giac [F]

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

input

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

output

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

3.1.60.9 Mupad [F(-1)]

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

input

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

output

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