3.1.28 \(\int \frac {\text {arccosh}(a x)^3}{x^2} \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 10, antiderivative size = 104 \[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=-\frac {\text {arccosh}(a x)^3}{x}+6 a \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-6 i a \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+6 i a \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+6 i a \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-6 i a \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right ) \]

output

-arccosh(a*x)^3/x+6*a*arccosh(a*x)^2*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2 
))-6*I*a*arccosh(a*x)*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+6*I* 
a*arccosh(a*x)*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+6*I*a*polylo 
g(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-6*I*a*polylog(3,I*(a*x+(a*x-1)^( 
1/2)*(a*x+1)^(1/2)))
 

3.1.28.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.23 \[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=-\frac {\text {arccosh}(a x)^3}{x}+3 i a \left (-\text {arccosh}(a x)^2 \left (\log \left (1-i e^{-\text {arccosh}(a x)}\right )-\log \left (1+i e^{-\text {arccosh}(a x)}\right )\right )-2 \text {arccosh}(a x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(a x)}\right )\right ) \]

input

Integrate[ArcCosh[a*x]^3/x^2,x]
 

output

-(ArcCosh[a*x]^3/x) + (3*I)*a*(-(ArcCosh[a*x]^2*(Log[1 - I/E^ArcCosh[a*x]] 
 - Log[1 + I/E^ArcCosh[a*x]])) - 2*ArcCosh[a*x]*(PolyLog[2, (-I)/E^ArcCosh 
[a*x]] - PolyLog[2, I/E^ArcCosh[a*x]]) - 2*PolyLog[3, (-I)/E^ArcCosh[a*x]] 
 + 2*PolyLog[3, I/E^ArcCosh[a*x]])
 

3.1.28.3 Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6298, 6362, 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.

\(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx\)

\(\Big \downarrow \) 6298

\(\displaystyle 3 a \int \frac {\text {arccosh}(a x)^2}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^3}{x}\)

\(\Big \downarrow \) 6362

\(\displaystyle 3 a \int \frac {\text {arccosh}(a x)^2}{a x}d\text {arccosh}(a x)-\frac {\text {arccosh}(a x)^3}{x}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcCosh[a*x]^3/x^2,x]
 

output

-(ArcCosh[a*x]^3/x) + 3*a*(2*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] + (2*I) 
*(-(ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]]) + PolyLog[3, (-I)*E^ArcC 
osh[a*x]]) - (2*I)*(-(ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]]) + PolyLog 
[3, I*E^ArcCosh[a*x]]))
 

3.1.28.3.1 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 6298

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

rule 6362

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 
_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ 
Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Subst 
[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, 
 e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte 
gerQ[m]
 

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]
 

3.1.28.4 Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{2}}d x\]

input

int(arccosh(a*x)^3/x^2,x)
 

output

int(arccosh(a*x)^3/x^2,x)
 

3.1.28.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arccosh(a*x)^3/x^2,x, algorithm="fricas")
 

output

integral(arccosh(a*x)^3/x^2, x)
 

3.1.28.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{2}}\, dx \]

input

integrate(acosh(a*x)**3/x**2,x)
 

output

Integral(acosh(a*x)**3/x**2, x)
 

3.1.28.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arccosh(a*x)^3/x^2,x, algorithm="maxima")
 

output

-log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x + integrate(3*(a^3*x^2 + sqrt( 
a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 
/(a^3*x^4 - a*x^2 + (a^2*x^3 - x)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
 

3.1.28.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arccosh(a*x)^3/x^2,x, algorithm="giac")
 

output

integrate(arccosh(a*x)^3/x^2, x)
 

3.1.28.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^2} \,d x \]

input

int(acosh(a*x)^3/x^2,x)
 

output

int(acosh(a*x)^3/x^2, x)