\(\int x^2 (a+b \text {sech}^{-1}(c x))^3 \, dx\) [43]

Optimal result

Integrand size = 14, antiderivative size = 242 \[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2 \arctan \left (e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \arctan \left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {i b^3 \operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {i b^3 \operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(c x)}\right )}{c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.82 \[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\frac {2 a^3 c^3 x^3-3 a^2 b c x \sqrt {\frac {1-c x}{1+c x}} (1+c x)+6 a^2 b c^3 x^3 \text {sech}^{-1}(c x)+3 i a^2 b \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )-6 a b^2 \left (c x+c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \text {sech}^{-1}(c x)-c^3 x^3 \text {sech}^{-1}(c x)^2-i \text {sech}^{-1}(c x) \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )+i \text {sech}^{-1}(c x) \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )\right )-b^3 \left (6 c x \text {sech}^{-1}(c x)+3 c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \text {sech}^{-1}(c x)^2-2 c^3 x^3 \text {sech}^{-1}(c x)^3-3 i \left (-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )\right )+\text {sech}^{-1}(c x)^2 \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x)^2 \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )+2 \text {sech}^{-1}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )-2 \text {sech}^{-1}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {sech}^{-1}(c x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {sech}^{-1}(c x)}\right )\right )\right )}{6 c^3} \] Input:

Integrate[x^2*(a + b*ArcSech[c*x])^3,x]
 

Output:

(2*a^3*c^3*x^3 - 3*a^2*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 6*a^2*b 
*c^3*x^3*ArcSech[c*x] + (3*I)*a^2*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + 
 c*x)]*(1 + c*x)] - 6*a*b^2*(c*x + c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) 
*ArcSech[c*x] - c^3*x^3*ArcSech[c*x]^2 - I*ArcSech[c*x]*Log[1 - I/E^ArcSec 
h[c*x]] + I*ArcSech[c*x]*Log[1 + I/E^ArcSech[c*x]] - I*PolyLog[2, (-I)/E^A 
rcSech[c*x]] + I*PolyLog[2, I/E^ArcSech[c*x]]) - b^3*(6*c*x*ArcSech[c*x] + 
 3*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*ArcSech[c*x]^2 - 2*c^3*x^3*ArcS 
ech[c*x]^3 - (3*I)*((-4*I)*ArcTan[Tanh[ArcSech[c*x]/2]] + ArcSech[c*x]^2*L 
og[1 - I/E^ArcSech[c*x]] - ArcSech[c*x]^2*Log[1 + I/E^ArcSech[c*x]] + 2*Ar 
cSech[c*x]*PolyLog[2, (-I)/E^ArcSech[c*x]] - 2*ArcSech[c*x]*PolyLog[2, I/E 
^ArcSech[c*x]] + 2*PolyLog[3, (-I)/E^ArcSech[c*x]] - 2*PolyLog[3, I/E^ArcS 
ech[c*x]])))/(6*c^3)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6839, 5974, 3042, 4674, 3042, 4257, 4668, 3011, 2720, 7143}

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.

\(\displaystyle \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx\)

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 5974

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4674

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4257

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[x^2*(a + b*ArcSech[c*x])^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, 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 4674

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

rule 5974

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

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])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{3}d x\]

Input:

int(x^2*(a+b*arcsech(c*x))^3,x)
 

Output:

int(x^2*(a+b*arcsech(c*x))^3,x)
 

Fricas [F]

\[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arcsech(c*x))^3,x, algorithm="fricas")
 

Output:

integral(b^3*x^2*arcsech(c*x)^3 + 3*a*b^2*x^2*arcsech(c*x)^2 + 3*a^2*b*x^2 
*arcsech(c*x) + a^3*x^2, x)
 

Sympy [F]

\[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx \] Input:

integrate(x**2*(a+b*asech(c*x))**3,x)
 

Output:

Integral(x**2*(a + b*asech(c*x))**3, x)
 

Maxima [F]

\[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arcsech(c*x))^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/3*a^3*x^3 + 1/2*(2*x^3*arcsech(c*x) - (sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^ 
2*x^2) - 1) + c^2) + arctan(sqrt(1/(c^2*x^2) - 1))/c^2)/c)*a^2*b + integra 
te(b^3*x^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^3 + 3*a*b^2* 
x^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2, x)
 

Giac [F]

\[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \] Input:

integrate(x^2*(a+b*arcsech(c*x))^3,x, algorithm="giac")
 

Output:

integrate((b*arcsech(c*x) + a)^3*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \] Input:

int(x^2*(a + b*acosh(1/(c*x)))^3,x)
 

Output:

int(x^2*(a + b*acosh(1/(c*x)))^3, x)
 

Reduce [F]

\[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=3 \left (\int \mathit {asech} \left (c x \right ) x^{2}d x \right ) a^{2} b +\left (\int \mathit {asech} \left (c x \right )^{3} x^{2}d x \right ) b^{3}+3 \left (\int \mathit {asech} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2}+\frac {a^{3} x^{3}}{3} \] Input:

int(x^2*(a+b*asech(c*x))^3,x)
 

Output:

(9*int(asech(c*x)*x**2,x)*a**2*b + 3*int(asech(c*x)**3*x**2,x)*b**3 + 9*in 
t(asech(c*x)**2*x**2,x)*a*b**2 + a**3*x**3)/3