\(\int \frac {e^{\text {sech}^{-1}(a x^p)}}{x} \, dx\) [68]

Optimal result

Integrand size = 12, antiderivative size = 85 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {x^{-p}}{a p}-\frac {\sqrt {-1+\frac {x^{-p}}{a}} \sqrt {1+\frac {x^{-p}}{a}}}{p}+\frac {\arctan \left (\sqrt {-1+\frac {x^{-p}}{a}} \sqrt {1+\frac {x^{-p}}{a}}\right )}{p} \] Output:

-1/a/p/(x^p)-(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2)/p+arctan((-1+1/a/(x^ 
p))^(1/2)*(1+1/a/(x^p))^(1/2))/p
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {i \left (-i x^{-p}-i \left (a+x^{-p}\right ) \sqrt {\frac {1-a x^p}{1+a x^p}}+a \log \left (-2 i a x^p+2 \sqrt {\frac {1-a x^p}{1+a x^p}} \left (1+a x^p\right )\right )\right )}{a p} \] Input:

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

Output:

((-I)*((-I)/x^p - I*(a + x^(-p))*Sqrt[(1 - a*x^p)/(1 + a*x^p)] + a*Log[(-2 
*I)*a*x^p + 2*Sqrt[(1 - a*x^p)/(1 + a*x^p)]*(1 + a*x^p)]))/(a*p)
 

Rubi [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6888, 791, 868, 773, 247, 223}

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.

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

\(\Big \downarrow \) 6888

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

\(\Big \downarrow \) 791

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

\(\Big \downarrow \) 868

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

\(\Big \downarrow \) 773

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 223

\(\displaystyle \frac {\sqrt {\frac {1}{a x^p+1}} \sqrt {a x^p+1} \left (x^p \left (-\sqrt {1-a^2 x^{-2 p}}\right )-a \arcsin \left (a x^p\right )\right )}{a p}-\frac {x^{-p}}{a p}\)

Input:

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

Output:

-(1/(a*p*x^p)) + (Sqrt[(1 + a*x^p)^(-1)]*Sqrt[1 + a*x^p]*(-(x^p*Sqrt[1 - a 
^2/x^(2*p)]) - a*ArcSin[a*x^p]))/(a*p)
 

Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 773

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

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 868

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) 
 Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[ 
{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]
 

rule 6888

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a \,x^{p}}{\sqrt {-a^{2} x^{2 p}+1}}\right ) a \,x^{p}+\operatorname {csgn}\left (a \right ) \sqrt {-a^{2} x^{2 p}+1}\right ) \operatorname {csgn}\left (a \right )}{\sqrt {-a^{2} x^{2 p}+1}}-\frac {x^{-p}}{a}}{p}\)	\(116\)
default	\(\frac {-\frac {\sqrt {-\frac {\left (a \,x^{p}-1\right ) x^{-p}}{a}}\, \sqrt {\frac {\left (1+a \,x^{p}\right ) x^{-p}}{a}}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a \,x^{p}}{\sqrt {-a^{2} x^{2 p}+1}}\right ) a \,x^{p}+\operatorname {csgn}\left (a \right ) \sqrt {-a^{2} x^{2 p}+1}\right ) \operatorname {csgn}\left (a \right )}{\sqrt {-a^{2} x^{2 p}+1}}-\frac {x^{-p}}{a}}{p}\)	\(116\)

Input:

int((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x,x,method=_RETUR 
NVERBOSE)
 

Output:

1/p*(-(-(a*x^p-1)/a/(x^p))^(1/2)*((1+a*x^p)/a/(x^p))^(1/2)*(arctan(csgn(a) 
*a*x^p/(-(x^p)^2*a^2+1)^(1/2))*a*x^p+csgn(a)*(-(x^p)^2*a^2+1)^(1/2))*csgn( 
a)/(-(x^p)^2*a^2+1)^(1/2)-1/a/(x^p))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^p\right )}}{x} \, dx=-\frac {a x^{p} \sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}} - a x^{p} \arctan \left (\sqrt {\frac {a x^{p} + 1}{a x^{p}}} \sqrt {-\frac {a x^{p} - 1}{a x^{p}}}\right ) + 1}{a p x^{p}} \] Input:

integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x,x, algori 
thm="fricas")
 

Output:

-(a*x^p*sqrt((a*x^p + 1)/(a*x^p))*sqrt(-(a*x^p - 1)/(a*x^p)) - a*x^p*arcta 
n(sqrt((a*x^p + 1)/(a*x^p))*sqrt(-(a*x^p - 1)/(a*x^p))) + 1)/(a*p*x^p)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x,x, algori 
thm="maxima")
 

Output:

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

Giac [F]

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

integrate((1/a/(x^p)+(-1+1/a/(x^p))^(1/2)*(1+1/a/(x^p))^(1/2))/x,x, algori 
thm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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