\(\int x \text {sech}^{-1}(a+b x)^3 \, dx\) [15]

Optimal result

Integrand size = 10, antiderivative size = 260 \[ \int x \text {sech}^{-1}(a+b x)^3 \, dx=-\frac {3 \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {3 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)^3+\frac {6 a \text {sech}^{-1}(a+b x)^2 \arctan \left (e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}+\frac {3 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 b^2}+\frac {6 i a \operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(a+b x)}\right )}{b^2} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.98 \[ \int x \text {sech}^{-1}(a+b x)^3 \, dx=\frac {-3 \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x) \text {sech}^{-1}(a+b x)^2-2 a (a+b x) \text {sech}^{-1}(a+b x)^3+(a+b x)^2 \text {sech}^{-1}(a+b x)^3+3 \text {sech}^{-1}(a+b x) \left (\text {sech}^{-1}(a+b x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(a+b x)}\right )\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a+b x)}\right )+6 i a \left (-\text {sech}^{-1}(a+b x)^2 \left (\log \left (1-i e^{-\text {sech}^{-1}(a+b x)}\right )-\log \left (1+i e^{-\text {sech}^{-1}(a+b x)}\right )\right )-2 \text {sech}^{-1}(a+b x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a+b x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{-\text {sech}^{-1}(a+b x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{-\text {sech}^{-1}(a+b x)}\right )\right )}{2 b^2} \] Input:

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

Output:

(-3*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x)*ArcSech[a + b*x]^2 
 - 2*a*(a + b*x)*ArcSech[a + b*x]^3 + (a + b*x)^2*ArcSech[a + b*x]^3 + 3*A 
rcSech[a + b*x]*(ArcSech[a + b*x] + 2*Log[1 + E^(-2*ArcSech[a + b*x])]) - 
3*PolyLog[2, -E^(-2*ArcSech[a + b*x])] + (6*I)*a*(-(ArcSech[a + b*x]^2*(Lo 
g[1 - I/E^ArcSech[a + b*x]] - Log[1 + I/E^ArcSech[a + b*x]])) - 2*ArcSech[ 
a + b*x]*(PolyLog[2, (-I)/E^ArcSech[a + b*x]] - PolyLog[2, I/E^ArcSech[a + 
 b*x]]) - 2*PolyLog[3, (-I)/E^ArcSech[a + b*x]] + 2*PolyLog[3, I/E^ArcSech 
[a + b*x]]))/(2*b^2)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6875, 25, 5991, 3042, 4678, 2009}

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.

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

\(\Big \downarrow \) 6875

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5991

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4678

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {1}{2} b^2 x^2 \text {sech}^{-1}(a+b x)^3+\frac {3}{2} \left (\frac {1}{3} a^2 \text {sech}^{-1}(a+b x)^3-4 a \text {sech}^{-1}(a+b x)^2 \arctan \left (e^{\text {sech}^{-1}(a+b x)}\right )+4 i a \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a+b x)}\right )-4 i a \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )-4 i a \operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a+b x)}\right )+4 i a \operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(a+b x)}\right )+\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^2+\text {sech}^{-1}(a+b x)^2-2 \text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )\right )}{b^2}\)

Input:

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

Output:

-((-1/2*(b^2*x^2*ArcSech[a + b*x]^3) + (3*(ArcSech[a + b*x]^2 + Sqrt[(1 - 
a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x]^2 + (a^2*ArcSech[a 
+ b*x]^3)/3 - 4*a*ArcSech[a + b*x]^2*ArcTan[E^ArcSech[a + b*x]] - 2*ArcSec 
h[a + b*x]*Log[1 + E^(2*ArcSech[a + b*x])] + (4*I)*a*ArcSech[a + b*x]*Poly 
Log[2, (-I)*E^ArcSech[a + b*x]] - (4*I)*a*ArcSech[a + b*x]*PolyLog[2, I*E^ 
ArcSech[a + b*x]] - PolyLog[2, -E^(2*ArcSech[a + b*x])] - (4*I)*a*PolyLog[ 
3, (-I)*E^ArcSech[a + b*x]] + (4*I)*a*PolyLog[3, I*E^ArcSech[a + b*x]]))/2 
)/b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4678

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
 

rule 5991

Int[((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*Sech[ 
(c_.) + (d_.)*(x_)])^(n_.)*Tanh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(-(e 
 + f*x)^m)*((a + b*Sech[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b 
*d*(n + 1)))   Int[(e + f*x)^(m - 1)*(a + b*Sech[c + d*x])^(n + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 6875

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Sech[x]*T 
anh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*x^2*log(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x + sqrt(b*x + a + 1)*s 
qrt(-b*x - a + 1)*a + b*x + a)^3 - integrate(1/2*(16*(b^3*x^4 + 3*a*b^2*x^ 
3 + (3*a^2*b - b)*x^2 + (a^3 - a)*x)*sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)* 
log(b*x + a)^3 + 16*(b^3*x^4 + 3*a*b^2*x^3 + (3*a^2*b - b)*x^2 + (a^3 - a) 
*x)*log(b*x + a)^3 + 3*(b^3*x^4 + 2*a*b^2*x^3 + (a^2*b - b)*x^2 + 4*(b^3*x 
^4 + 3*a*b^2*x^3 + (3*a^2*b - b)*x^2 + (a^3 - a)*x)*log(b*x + a) + (2*(b^3 
*x^4 + 3*a*b^2*x^3 + (3*a^2*b - b)*x^2 + (a^3 - a)*x)*sqrt(b*x + a + 1)*lo 
g(b*x + a) + (2*b^3*x^4 + 4*a*b^2*x^3 + (2*a^2*b - b)*x^2 + 2*(b^3*x^4 + 3 
*a*b^2*x^3 + (3*a^2*b - b)*x^2 + (a^3 - a)*x)*log(b*x + a))*sqrt(b*x + a + 
 1))*sqrt(-b*x - a + 1))*log(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x + sq 
rt(b*x + a + 1)*sqrt(-b*x - a + 1)*a + b*x + a)^2 - 24*((b^3*x^4 + 3*a*b^2 
*x^3 + (3*a^2*b - b)*x^2 + (a^3 - a)*x)*sqrt(b*x + a + 1)*sqrt(-b*x - a + 
1)*log(b*x + a)^2 + (b^3*x^4 + 3*a*b^2*x^3 + (3*a^2*b - b)*x^2 + (a^3 - a) 
*x)*log(b*x + a)^2)*log(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x + sqrt(b* 
x + a + 1)*sqrt(-b*x - a + 1)*a + b*x + a))/(b^3*x^3 + 3*a*b^2*x^2 + a^3 + 
 (b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b - b)*x - a)*sqrt(b*x + a + 1)*sqr 
t(-b*x - a + 1) + (3*a^2*b - b)*x - a), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(asech(a + b*x)**3*x,x)