3.1.47 \(\int e^{\text {sech}^{-1}(a x^2)} x^5 \, dx\) [47]

3.1.47.1 Optimal result

Integrand size = 12, antiderivative size = 58 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {x^4}{12 a}+\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6-\frac {\sqrt {1-a x^2}}{6 a^3 \sqrt {\frac {1}{1+a x^2}}} \]

output

1/12*x^4/a+1/6*(1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^6-1/6*(-a*x 
^2+1)^(1/2)/a^3/(1/(a*x^2+1))^(1/2)
 

3.1.47.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {x^4}{4 a}+\frac {\left (-1+a x^2\right ) \sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )^2}{6 a^3} \]

input

Integrate[E^ArcSech[a*x^2]*x^5,x]
 

output

x^4/(4*a) + ((-1 + a*x^2)*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2)^2)/(6* 
a^3)
 

3.1.47.3 Rubi [A] (warning: unable to verify)

Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6889, 15, 335, 793}

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.

\(\displaystyle \int x^5 e^{\text {sech}^{-1}\left (a x^2\right )} \, dx\)

\(\Big \downarrow \) 6889

\(\displaystyle \frac {\int x^3dx}{3 a}+\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^3}{\sqrt {1-a x^2} \sqrt {a x^2+1}}dx}{3 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^3}{\sqrt {1-a x^2} \sqrt {a x^2+1}}dx}{3 a}+\frac {x^4}{12 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )}\)

\(\Big \downarrow \) 335

\(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^3}{\sqrt {1-a^2 x^4}}dx}{3 a}+\frac {x^4}{12 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )}\)

\(\Big \downarrow \) 793

\(\displaystyle -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{6 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )}\)

input

Int[E^ArcSech[a*x^2]*x^5,x]
 

output

x^4/(12*a) + (E^ArcSech[a*x^2]*x^6)/6 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a 
*x^2]*Sqrt[1 - a^2*x^4])/(6*a^3)
 

3.1.47.3.1 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 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 793

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

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]
 

3.1.47.4 Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (x^{4} a^{2}-1\right )}{6 a^{2}}+\frac {x^{4}}{4 a}\)	\(60\)

input

int((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x,method=_RETURNVERB 
OSE)
 

output

1/6*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(a^2*x^4-1)/a^2+1 
/4*x^4/a
 

3.1.47.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {3 \, a x^{4} + 2 \, {\left (a^{2} x^{6} - x^{2}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}}}{12 \, a^{2}} \]

input

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x, algorithm=" 
fricas")
 

output

1/12*(3*a*x^4 + 2*(a^2*x^6 - x^2)*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 
 1)/(a*x^2)))/a^2
 

3.1.47.6 Sympy [F]

\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {\int x^{3}\, dx + \int a x^{5} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]

input

integrate((1/a/x**2+(1/a/x**2-1)**(1/2)*(1/a/x**2+1)**(1/2))*x**5,x)
 

output

(Integral(x**3, x) + Integral(a*x**5*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x 
**2)), x))/a
 

3.1.47.7 Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {x^{4}}{4 \, a} + \frac {{\left (a^{2} x^{4} - 1\right )} \sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}}{6 \, a^{3}} \]

input

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x, algorithm=" 
maxima")
 

output

1/4*x^4/a + 1/6*(a^2*x^4 - 1)*sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/a^3
 

3.1.47.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (69) = 138\).

Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.28 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {{\left (\sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (a^{2} x^{2} + a\right )} {\left (\frac {2 \, {\left (a^{2} x^{2} + a\right )}}{a^{4}} - \frac {7}{a^{3}}\right )} + \frac {9}{a^{2}}\right )} + \frac {6 \, \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right )}{a}\right )} a - \frac {3 \, {\left (2 \, a^{2} \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) - \sqrt {a^{2} x^{2} + a} {\left (a^{2} x^{2} - 2 \, a\right )} \sqrt {-a^{2} x^{2} + a}\right )}}{a^{2}} + \frac {3 \, {\left ({\left (a^{2} x^{2} + a\right )}^{2} - 2 \, {\left (a^{2} x^{2} + a\right )} a\right )}}{a^{2}}}{12 \, a^{3}} \]

input

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x, algorithm=" 
giac")
 

output

1/12*((sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2 + a)*((a^2*x^2 + a)*(2*(a^2*x^2 + a 
)/a^4 - 7/a^3) + 9/a^2) + 6*arcsin(1/2*sqrt(2)*sqrt(a^2*x^2 + a)/sqrt(a))/ 
a)*a - 3*(2*a^2*arcsin(1/2*sqrt(2)*sqrt(a^2*x^2 + a)/sqrt(a)) - sqrt(a^2*x 
^2 + a)*(a^2*x^2 - 2*a)*sqrt(-a^2*x^2 + a))/a^2 + 3*((a^2*x^2 + a)^2 - 2*( 
a^2*x^2 + a)*a)/a^2)/a^3
 

3.1.47.9 Mupad [B] (verification not implemented)

Time = 5.47 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\sqrt {\frac {1}{a\,x^2}-1}\,\left (\frac {x^6\,\sqrt {\frac {1}{a\,x^2}+1}}{6}-\frac {x^2\,\sqrt {\frac {1}{a\,x^2}+1}}{6\,a^2}\right )+\frac {x^4}{4\,a} \]

input

int(x^5*((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2)),x)
 

output

(1/(a*x^2) - 1)^(1/2)*((x^6*(1/(a*x^2) + 1)^(1/2))/6 - (x^2*(1/(a*x^2) + 1 
)^(1/2))/(6*a^2)) + x^4/(4*a)