\(\int e^{\text {sech}^{-1}(a x^2)} x^2 \, dx\) [50]

Optimal result

Integrand size = 12, antiderivative size = 67 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx=\frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{3 a^{3/2}} \]

[Out]

2/3*x/a+1/3*(1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^3+2/3*EllipticF(x*a^(1/2),I)*(1/(a*x^2+1))^(1/2)*(
a*x^2+1)^(1/2)/a^(3/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, 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 = {6470, 8, 254, 227} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx=\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{3 a^{3/2}}+\frac {1}{3} x^3 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {2 x}{3 a} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 227

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

Rule 254

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

Rule 6470

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ArcSech[a*x^p]/(m + 1)), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \int 1 \, dx}{3 a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{3 a} \\ & = \frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx}{3 a} \\ & = \frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {a} x\right ),-1\right )}{3 a^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.67 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx=-\frac {2 \sqrt {2} e^{-\text {sech}^{-1}\left (a x^2\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^2\right )}}\right )^{3/2} x \left (-1-2 e^{2 \text {sech}^{-1}\left (a x^2\right )}+\left (1+e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-e^{2 \text {sech}^{-1}\left (a x^2\right )}\right )\right )}{3 a \sqrt {a x^2}} \]

[In]

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

[Out]

(-2*Sqrt[2]*(E^ArcSech[a*x^2]/(1 + E^(2*ArcSech[a*x^2])))^(3/2)*x*(-1 - 2*E^(2*ArcSech[a*x^2]) + (1 + E^(2*Arc
Sech[a*x^2]))^(3/2)*Hypergeometric2F1[1/4, 1/2, 5/4, -E^(2*ArcSech[a*x^2])]))/(3*a*E^ArcSech[a*x^2]*Sqrt[a*x^2
])

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{\frac {5}{2}} x^{5}-2 \operatorname {EllipticF}\left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}-x \sqrt {a}\right )}{3 \left (x^{4} a^{2}-1\right ) \sqrt {a}}+\frac {x}{a}\)	\(102\)

[In]

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

[Out]

1/3*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(a^(5/2)*x^5-2*EllipticF(x*a^(1/2),I)*(-a*x^2+1)^(1/2
)*(a*x^2+1)^(1/2)-x*a^(1/2))/(a^2*x^4-1)/a^(1/2)+x/a

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx=\frac {a x^{3} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 3 \, x + \frac {2 i \, F(\arcsin \left (\frac {1}{\sqrt {a} x}\right )\,|\,-1)}{\sqrt {a}}}{3 \, a} \]

[In]

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

[Out]

1/3*(a*x^3*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) + 3*x + 2*I*elliptic_f(arcsin(1/(sqrt(a)*x)),
-1)/sqrt(a))/a

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,2,1,1,1]%%%}+%%%{1,[0,0,0,0,2]%%%} / %%%{1,[0,0,0,0
,3]%%%} Err

Mupad [F(-1)]

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

[In]

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

[Out]

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