3.1.3 \(\int x^2 \text {sech}^{-1}(a x)^2 \, dx\) [3]

3.1.3.1 Optimal result

Integrand size = 10, antiderivative size = 117 \[ \int x^2 \text {sech}^{-1}(a x)^2 \, dx=-\frac {x}{3 a^2}-\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \text {sech}^{-1}(a x)^2-\frac {2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}+\frac {i \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3}-\frac {i \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{3 a^3} \]

output

-1/3*x/a^2+1/3*x^3*arcsech(a*x)^2-2/3*arcsech(a*x)*arctan(1/a/x+(1/a/x-1)^ 
(1/2)*(1+1/a/x)^(1/2))/a^3+1/3*I*polylog(2,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/ 
a/x)^(1/2)))/a^3-1/3*I*polylog(2,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)) 
)/a^3-1/3*x*(a*x+1)*arcsech(a*x)*((-a*x+1)/(a*x+1))^(1/2)/a^2
 

3.1.3.2 Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int x^2 \text {sech}^{-1}(a x)^2 \, dx=\frac {-a x-a x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)+a^3 x^3 \text {sech}^{-1}(a x)^2+i \text {sech}^{-1}(a x) \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-i \text {sech}^{-1}(a x) \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a x)}\right )}{3 a^3} \]

input

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

output

(-(a*x) - a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] + a^3*x^3*A 
rcSech[a*x]^2 + I*ArcSech[a*x]*Log[1 - I/E^ArcSech[a*x]] - I*ArcSech[a*x]* 
Log[1 + I/E^ArcSech[a*x]] + I*PolyLog[2, (-I)/E^ArcSech[a*x]] - I*PolyLog[ 
2, I/E^ArcSech[a*x]])/(3*a^3)
 

3.1.3.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6839, 5941, 3042, 4673, 3042, 4668, 2715, 2838}

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.

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

\(\Big \downarrow \) 6839

\(\displaystyle -\frac {\int a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)}{a^3}\)

\(\Big \downarrow \) 5941

\(\displaystyle -\frac {\frac {2}{3} \int a^3 x^3 \text {sech}^{-1}(a x)d\text {sech}^{-1}(a x)-\frac {1}{3} a^3 x^3 \text {sech}^{-1}(a x)^2}{a^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{3} a^3 x^3 \text {sech}^{-1}(a x)^2+\frac {2}{3} \int \text {sech}^{-1}(a x) \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(a x)}{a^3}\)

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{3} a^3 x^3 \text {sech}^{-1}(a x)^2+\frac {2}{3} \left (\frac {1}{2} \int \text {sech}^{-1}(a x) \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)+\frac {a x}{2}+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)\right )}{a^3}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {-\frac {1}{3} a^3 x^3 \text {sech}^{-1}(a x)^2+\frac {2}{3} \left (\frac {1}{2} \left (-i \int \log \left (1-i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)+i \int \log \left (1+i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)+2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )+\frac {a x}{2}+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)\right )}{a^3}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {1}{3} a^3 x^3 \text {sech}^{-1}(a x)^2+\frac {2}{3} \left (\frac {1}{2} \left (-i \int e^{-\text {sech}^{-1}(a x)} \log \left (1-i e^{\text {sech}^{-1}(a x)}\right )de^{\text {sech}^{-1}(a x)}+i \int e^{-\text {sech}^{-1}(a x)} \log \left (1+i e^{\text {sech}^{-1}(a x)}\right )de^{\text {sech}^{-1}(a x)}+2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )+\frac {a x}{2}+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)\right )}{a^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {-\frac {1}{3} a^3 x^3 \text {sech}^{-1}(a x)^2+\frac {2}{3} \left (\frac {1}{2} \left (2 \text {sech}^{-1}(a x) \arctan \left (e^{\text {sech}^{-1}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )\right )+\frac {a x}{2}+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)\right )}{a^3}\)

input

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

output

-((-1/3*(a^3*x^3*ArcSech[a*x]^2) + (2*((a*x)/2 + (a*x*Sqrt[(1 - a*x)/(1 + 
a*x)]*(1 + a*x)*ArcSech[a*x])/2 + (2*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]] - 
 I*PolyLog[2, (-I)*E^ArcSech[a*x]] + I*PolyLog[2, I*E^ArcSech[a*x]])/2))/3 
)/a^3)
 

3.1.3.3.1 Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 5941

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_) 
^(n_.)]^(q_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p 
)), x] + Simp[(m - n + 1)/(b*n*p)   Int[x^(m - n)*Sech[a + b*x^n]^p, x], x] 
 /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && 
EqQ[q, 1]
 

rule 6839

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A 
rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

3.1.3.4 Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.97

method	result	size
derivativedivides	\(\frac {\frac {\left (a^{2} x^{2} \operatorname {arcsech}\left (a x \right )^{2}-\operatorname {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x -1\right ) a x}{3}+\frac {i \operatorname {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}-\frac {i \operatorname {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}}{a^{3}}\)	\(230\)
default	\(\frac {\frac {\left (a^{2} x^{2} \operatorname {arcsech}\left (a x \right )^{2}-\operatorname {arcsech}\left (a x \right ) \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, a x -1\right ) a x}{3}+\frac {i \operatorname {arcsech}\left (a x \right ) \ln \left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}-\frac {i \operatorname {arcsech}\left (a x \right ) \ln \left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )\right )}{3}}{a^{3}}\)	\(230\)

input

int(x^2*arcsech(a*x)^2,x,method=_RETURNVERBOSE)
 

output

1/a^3*(1/3*(a^2*x^2*arcsech(a*x)^2-arcsech(a*x)*(-(a*x-1)/a/x)^(1/2)*((a*x 
+1)/a/x)^(1/2)*a*x-1)*a*x+1/3*I*arcsech(a*x)*ln(1+I*(1/a/x+(1/a/x-1)^(1/2) 
*(1+1/a/x)^(1/2)))-1/3*I*arcsech(a*x)*ln(1-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a 
/x)^(1/2)))+1/3*I*dilog(1+I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))-1/3*I 
*dilog(1-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))))
 

3.1.3.5 Fricas [F]

\[ \int x^2 \text {sech}^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arsech}\left (a x\right )^{2} \,d x } \]

input

integrate(x^2*arcsech(a*x)^2,x, algorithm="fricas")
 

output

integral(x^2*arcsech(a*x)^2, x)
 

3.1.3.6 Sympy [F]

\[ \int x^2 \text {sech}^{-1}(a x)^2 \, dx=\int x^{2} \operatorname {asech}^{2}{\left (a x \right )}\, dx \]

input

integrate(x**2*asech(a*x)**2,x)
 

output

Integral(x**2*asech(a*x)**2, x)
 

3.1.3.7 Maxima [F]

\[ \int x^2 \text {sech}^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arsech}\left (a x\right )^{2} \,d x } \]

input

integrate(x^2*arcsech(a*x)^2,x, algorithm="maxima")
 

output

integrate(x^2*arcsech(a*x)^2, x)
 

3.1.3.8 Giac [F]

\[ \int x^2 \text {sech}^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arsech}\left (a x\right )^{2} \,d x } \]

input

integrate(x^2*arcsech(a*x)^2,x, algorithm="giac")
 

output

integrate(x^2*arcsech(a*x)^2, x)
 

3.1.3.9 Mupad [F(-1)]

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

input

int(x^2*acosh(1/(a*x))^2,x)
 

output

int(x^2*acosh(1/(a*x))^2, x)