3.1.45 \(\int (a+b \text {sech}^{-1}(c x))^3 \, dx\) [45]

3.1.45.1 Optimal result

Integrand size = 10, antiderivative size = 140 \[ \int \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=x \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {6 b \left (a+b \text {sech}^{-1}(c x)\right )^2 \arctan \left (e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{c}-\frac {6 i b^3 \operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(c x)}\right )}{c}+\frac {6 i b^3 \operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(c x)}\right )}{c} \]

output

x*(a+b*arcsech(c*x))^3-6*b*(a+b*arcsech(c*x))^2*arctan(1/c/x+(-1+1/c/x)^(1 
/2)*(1+1/c/x)^(1/2))/c+6*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-6*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-6*I*b^3*polylog(3,-I*(1/c/x+(-1+ 
1/c/x)^(1/2)*(1+1/c/x)^(1/2)))/c+6*I*b^3*polylog(3,I*(1/c/x+(-1+1/c/x)^(1/ 
2)*(1+1/c/x)^(1/2)))/c
 

3.1.45.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(282\) vs. \(2(140)=280\).

Time = 0.57 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.01 \[ \int \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx=a^3 x+3 a^2 b x \text {sech}^{-1}(c x)-\frac {3 a^2 b \arctan \left (\frac {c x \sqrt {\frac {1-c x}{1+c x}}}{-1+c x}\right )}{c}+\frac {3 i a b^2 \left (\text {sech}^{-1}(c x) \left (-i c x \text {sech}^{-1}(c x)+2 \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-2 \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )-2 \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )\right )}{c}+\frac {b^3 \left (c x \text {sech}^{-1}(c x)^3-3 i \left (-\text {sech}^{-1}(c x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )-2 \text {sech}^{-1}(c x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )\right )-2 \left (\operatorname {PolyLog}\left (3,-i e^{-\text {sech}^{-1}(c x)}\right )-\operatorname {PolyLog}\left (3,i e^{-\text {sech}^{-1}(c x)}\right )\right )\right )\right )}{c} \]

input

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

output

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

3.1.45.3 Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6833, 5974, 3042, 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.

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

\(\Big \downarrow \) 6833

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

\(\Big \downarrow \) 5974

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

-((-(c*x*(a + b*ArcSech[c*x])^3) + 3*b*(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^ArcSech[c*x]])))/c)
 

3.1.45.3.1 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 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]
 

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]
 

3.1.45.4 Maple [F]

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

input

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

output

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

3.1.45.5 Fricas [F]

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

input

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

output

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

3.1.45.6 Sympy [F]

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

input

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

output

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

3.1.45.7 Maxima [F]

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

input

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

output

b^3*x*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^3 + a^3*x + 3*(c*x*arcsech(c*x 
) - arctan(sqrt(1/(c^2*x^2) - 1)))*a^2*b/c - integrate(-(b^3*log(c)^3 - 3* 
a*b^2*log(c)^2 - (b^3*c^2*x^2 - b^3)*log(x)^3 - (b^3*c^2*log(c)^3 - 3*a*b^ 
2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 - (b^3*c^2*log(c) - a*b^2*c^2) 
*x^2 + (b^3*log(c) - a*b^2 - (b^3*c^2*(log(c) + 1) - a*b^2*c^2)*x^2 - (b^3 
*c^2*x^2 - b^3)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^3*c^2*x^2 - b^3) 
*log(x))*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)^2 + 3*(b^3*log(c) - a*b^2 - 
 (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x)^2 + (b^3*log(c)^3 - 3*a*b^2*log( 
c)^2 - (b^3*c^2*x^2 - b^3)*log(x)^3 - (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log( 
c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 - (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log( 
x)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) - (b^3*c^2*log(c)^2 - 2*a*b^2*c^2* 
log(c))*x^2)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - 3*(b^3*log(c)^2 - 2*a* 
b^2*log(c) - (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2 - (b^3*c^2*x^2 - 
b^3)*log(x)^2 + (b^3*log(c)^2 - 2*a*b^2*log(c) - (b^3*c^2*log(c)^2 - 2*a*b 
^2*c^2*log(c))*x^2 - (b^3*c^2*x^2 - b^3)*log(x)^2 + 2*(b^3*log(c) - a*b^2 
- (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) + 
 2*(b^3*log(c) - a*b^2 - (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x))*log(sqr 
t(c*x + 1)*sqrt(-c*x + 1) + 1) + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) - (b^3*c 
^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2)*log(x))/(c^2*x^2 + (c^2*x^2 - 1)*sq 
rt(c*x + 1)*sqrt(-c*x + 1) - 1), x)
 

3.1.45.8 Giac [F]

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

input

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

output

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

3.1.45.9 Mupad [F(-1)]

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

input

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

output

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