3.1.34 \(\int x^2 (a+b \text {sech}^{-1}(c x))^2 \, dx\) [34]

3.1.34.1 Optimal result

Integrand size = 14, antiderivative size = 140 \[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=-\frac {b^2 x}{3 c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right ) \arctan \left (e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3} \]

output

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

3.1.34.2 Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=\frac {1}{3} \left (a^2 x^3+a b \left (2 x^3 \text {sech}^{-1}(c x)+\frac {\sqrt {\frac {1-c x}{1+c x}} \left (c x-c^3 x^3+2 \sqrt {1-c^2 x^2} \arctan \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c^3 (-1+c x)}\right )+\frac {b^2 \left (-c x-c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \text {sech}^{-1}(c x)+c^3 x^3 \text {sech}^{-1}(c x)^2+i \text {sech}^{-1}(c x) \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-i \text {sech}^{-1}(c x) \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )\right )}{c^3}\right ) \]

input

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

output

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

3.1.34.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6839, 5974, 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.

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

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 5974

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.1.34.3.1 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 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])
 

3.1.34.4 Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.34

method	result	size
parts	\(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )^{2}-\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1\right ) c x}{3}+\frac {i \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \operatorname {arcsech}\left (c x \right )}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\)	\(328\)
derivativedivides	\(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )^{2}-\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1\right ) c x}{3}+\frac {i \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arcsech}\left (c x \right )}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\)	\(329\)
default	\(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arcsech}\left (c x \right )^{2}-\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1\right ) c x}{3}+\frac {i \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}\right )+2 a b \left (\frac {c^{3} x^{3} \operatorname {arcsech}\left (c x \right )}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\)	\(329\)

input

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

output

1/3*a^2*x^3+b^2/c^3*(1/3*(c^2*x^2*arcsech(c*x)^2-arcsech(c*x)*(-(c*x-1)/c/ 
x)^(1/2)*((c*x+1)/c/x)^(1/2)*c*x-1)*c*x+1/3*I*arcsech(c*x)*ln(1+I*(1/c/x+( 
-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-1/3*I*arcsech(c*x)*ln(1-I*(1/c/x+(-1+1/c 
/x)^(1/2)*(1+1/c/x)^(1/2)))+1/3*I*dilog(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c 
/x)^(1/2)))-1/3*I*dilog(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))))+2*a 
*b/c^3*(1/3*c^3*x^3*arcsech(c*x)+1/6*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x 
)^(1/2)*(-c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/(-c^2*x^2+1)^(1/2))
 

3.1.34.5 Fricas [F]

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

input

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

output

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

3.1.34.6 Sympy [F]

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

input

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

output

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

3.1.34.7 Maxima [F]

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

input

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

output

1/3*a^2*x^3 + 1/3*(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*b + b^2*integ 
rate(x^2*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^2, x)
 

3.1.34.8 Giac [F]

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

input

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

output

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

3.1.34.9 Mupad [F(-1)]

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

input

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

output

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