\(\int e^{\text {sech}^{-1}(a x^2)} \, dx\) [58]

Optimal result

Integrand size = 8, antiderivative size = 185 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=-\frac {1}{a x}-\frac {\sqrt {1+a x^2} \sqrt {1-a^2 x^4} \sqrt {-1+\frac {2}{1+a x^2}}}{a x \sqrt {1-a x^2}}-\frac {2 \sqrt {1+a x^2} \sqrt {-1+\frac {2}{1+a x^2}} E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a} \sqrt {1-a x^2}}+\frac {2 \sqrt {1+a x^2} \sqrt {-1+\frac {2}{1+a x^2}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{\sqrt {a} \sqrt {1-a x^2}} \] Output:

-1/a/x-(a*x^2+1)^(1/2)*(-a^2*x^4+1)^(1/2)*(-1+2/(a*x^2+1))^(1/2)/a/x/(-a*x 
^2+1)^(1/2)-2*(a*x^2+1)^(1/2)*(-1+2/(a*x^2+1))^(1/2)*EllipticE(a^(1/2)*x,I 
)/a^(1/2)/(-a*x^2+1)^(1/2)+2*(a*x^2+1)^(1/2)*(-1+2/(a*x^2+1))^(1/2)*Ellipt 
icF(a^(1/2)*x,I)/a^(1/2)/(-a*x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.73 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=-\frac {1}{a x}+\left (-\frac {1}{a x}-x\right ) \sqrt {\frac {1-a x^2}{1+a x^2}}-\frac {2 i \sqrt {\frac {1-a x^2}{1+a x^2}} \sqrt {1-a^2 x^4} \left (E\left (\left .i \text {arcsinh}\left (\sqrt {-a} x\right )\right |-1\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-a} x\right ),-1\right )\right )}{\sqrt {-a} \left (-1+a x^2\right )} \] Input:

Integrate[E^ArcSech[a*x^2],x]
 

Output:

-(1/(a*x)) + (-(1/(a*x)) - x)*Sqrt[(1 - a*x^2)/(1 + a*x^2)] - ((2*I)*Sqrt[ 
(1 - a*x^2)/(1 + a*x^2)]*Sqrt[1 - a^2*x^4]*(EllipticE[I*ArcSinh[Sqrt[-a]*x 
], -1] - EllipticF[I*ArcSinh[Sqrt[-a]*x], -1]))/(Sqrt[-a]*(-1 + a*x^2))
 

Rubi [A] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.58, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6884, 15, 335, 847, 836, 762, 1388, 327}

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.

\(\displaystyle \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx\)

\(\Big \downarrow \) 6884

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 335

\(\displaystyle \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {1}{x^2 \sqrt {1-a^2 x^4}}dx}{a}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x}\)

\(\Big \downarrow \) 847

\(\displaystyle \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (a^2 \left (-\int \frac {x^2}{\sqrt {1-a^2 x^4}}dx\right )-\frac {\sqrt {1-a^2 x^4}}{x}\right )}{a}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (-\left (a^2 \left (\frac {\int \frac {a x^2+1}{\sqrt {1-a^2 x^4}}dx}{a}-\frac {\int \frac {1}{\sqrt {1-a^2 x^4}}dx}{a}\right )\right )-\frac {\sqrt {1-a^2 x^4}}{x}\right )}{a}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (-\left (a^2 \left (\frac {\int \frac {a x^2+1}{\sqrt {1-a^2 x^4}}dx}{a}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{a^{3/2}}\right )\right )-\frac {\sqrt {1-a^2 x^4}}{x}\right )}{a}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (-\left (a^2 \left (\frac {\int \frac {\sqrt {a x^2+1}}{\sqrt {1-a x^2}}dx}{a}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{a^{3/2}}\right )\right )-\frac {\sqrt {1-a^2 x^4}}{x}\right )}{a}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (-\frac {\sqrt {1-a^2 x^4}}{x}-\left (a^2 \left (\frac {E\left (\left .\arcsin \left (\sqrt {a} x\right )\right |-1\right )}{a^{3/2}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{a^{3/2}}\right )\right )\right )}{a}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x}\)

Input:

Int[E^ArcSech[a*x^2],x]
 

Output:

-2/(a*x) + E^ArcSech[a*x^2]*x + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]* 
(-(Sqrt[1 - a^2*x^4]/x) - a^2*(EllipticE[ArcSin[Sqrt[a]*x], -1]/a^(3/2) - 
EllipticF[ArcSin[Sqrt[a]*x], -1]/a^(3/2))))/a
 

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 327

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

rule 335

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p 
_.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] 
))
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 847

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

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 6884

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71

method	result	size
default	\(-\frac {1}{a x}-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{2} x^{4}+2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \operatorname {EllipticF}\left (\sqrt {a}\, x , i\right ) \sqrt {a}-2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \operatorname {EllipticE}\left (\sqrt {a}\, x , i\right ) \sqrt {a}-1\right )}{a^{2} x^{4}-1}\)	\(132\)

Input:

int(1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/a/x-(-(a*x^2-1)/a/x^2)^(1/2)*x*((a*x^2+1)/a/x^2)^(1/2)*(a^2*x^4+2*(-a*x 
^2+1)^(1/2)*(a*x^2+1)^(1/2)*x*EllipticF(a^(1/2)*x,I)*a^(1/2)-2*(-a*x^2+1)^ 
(1/2)*(a*x^2+1)^(1/2)*x*EllipticE(a^(1/2)*x,I)*a^(1/2)-1)/(a^2*x^4-1)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

(Integral(x**(-2), x) + Integral(a*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x** 
2)), x))/a
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx=\frac {\sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}-2 \left (\int \frac {\sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}}{a^{2} x^{6}-x^{2}}d x \right ) x -1}{a x} \] Input:

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

Output:

(sqrt(a*x**2 + 1)*sqrt( - a*x**2 + 1) - 2*int((sqrt(a*x**2 + 1)*sqrt( - a* 
x**2 + 1))/(a**2*x**6 - x**2),x)*x - 1)/(a*x)