\(\int (a+b \text {sech}^{-1}(c x))^2 \, dx\) [36]

Optimal result

Integrand size = 10, antiderivative size = 78 \[ \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=x \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {4 b \left (a+b \text {sech}^{-1}(c x)\right ) \arctan \left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{c} \] Output:

x*(a+b*arcsech(c*x))^2-4*b*(a+b*arcsech(c*x))*arctan(1/c/x+(-1+1/c/x)^(1/2 
)*(1+1/c/x)^(1/2))/c+2*I*b^2*polylog(2,-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x 
)^(1/2)))/c-2*I*b^2*polylog(2,I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))/ 
c
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.62 \[ \int \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx=a^2 x+\frac {2 a b \left (c x \text {sech}^{-1}(c x)-2 \arctan \left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )\right )\right )}{c}+\frac {i b^2 \left (\text {sech}^{-1}(c x) \left (-i c x \text {sech}^{-1}(c x)+2 \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )-2 \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )\right )}{c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6833, 5974, 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.

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

\(\Big \downarrow \) 6833

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

\(\Big \downarrow \) 5974

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((-(c*x*(a + b*ArcSech[c*x])^2) + 2*b*(2*(a + b*ArcSech[c*x])*ArcTan[E^Ar 
cSech[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSech[c*x]] + I*b*PolyLog[2, I*E^Arc 
Sech[c*x]]))/c)
 

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 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 6833

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1)   S 
ubst[Int[(a + b*x)^n*Sech[x]*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, 
 b, c, n}, x] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.97

method	result	size
derivativedivides	\(\frac {c x \,a^{2}+b^{2} \left (\operatorname {arcsech}\left (c x \right )^{2} c x +2 i \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )-2 i \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )\right )+2 a b \left (c x \,\operatorname {arcsech}\left (c x \right )-\arctan \left (\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{c}\)	\(232\)
default	\(\frac {c x \,a^{2}+b^{2} \left (\operatorname {arcsech}\left (c x \right )^{2} c x +2 i \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )-2 i \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )\right )+2 a b \left (c x \,\operatorname {arcsech}\left (c x \right )-\arctan \left (\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{c}\)	\(232\)
parts	\(a^{2} x +\frac {b^{2} \left (\operatorname {arcsech}\left (c x \right )^{2} c x +2 i \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )-2 i \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )\right )}{c}+2 b x a \,\operatorname {arcsech}\left (c x \right )-\frac {2 a b \arctan \left (\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{c}\)	\(232\)

Input:

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

Output:

1/c*(c*x*a^2+b^2*(arcsech(c*x)^2*c*x+2*I*arcsech(c*x)*ln(1+I*(1/c/x+(-1+1/ 
c/x)^(1/2)*(1+1/c/x)^(1/2)))-2*I*arcsech(c*x)*ln(1-I*(1/c/x+(-1+1/c/x)^(1/ 
2)*(1+1/c/x)^(1/2)))+2*I*dilog(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2) 
))-2*I*dilog(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))))+2*a*b*(c*x*arc 
sech(c*x)-arctan((-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

(x*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^2 - integrate(-(c^2*x^2*log(c)^2 
+ (c^2*x^2 - 1)*log(x)^2 + (c^2*x^2*log(c)^2 + (c^2*x^2 - 1)*log(x)^2 - lo 
g(c)^2 + 2*(c^2*x^2*log(c) - log(c))*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) 
- 2*(c^2*x^2*log(c) + (c^2*x^2*(log(c) + 1) + (c^2*x^2 - 1)*log(x) - log(c 
))*sqrt(c*x + 1)*sqrt(-c*x + 1) + (c^2*x^2 - 1)*log(x) - log(c))*log(sqrt( 
c*x + 1)*sqrt(-c*x + 1) + 1) - log(c)^2 + 2*(c^2*x^2*log(c) - log(c))*log( 
x))/(c^2*x^2 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(-c*x + 1) - 1), x))*b^2 + 
a^2*x + 2*(c*x*arcsech(c*x) - arctan(sqrt(1/(c^2*x^2) - 1)))*a*b/c
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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