3.1.30 \(\int \frac {\text {arccosh}(a x)^3}{x^4} \, dx\) [30]

3.1.30.1 Optimal result

Integrand size = 10, antiderivative size = 183 \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\frac {a^2 \text {arccosh}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x^2}-\frac {\text {arccosh}(a x)^3}{3 x^3}+a^3 \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )-a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i a^3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+i a^3 \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-i a^3 \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right ) \]

output

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

3.1.30.2 Mathematica [A] (warning: unable to verify)

Time = 0.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (\frac {6 a^2 \text {arccosh}(a x)}{x}+\frac {3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^2}{x^2}-\frac {2 \text {arccosh}(a x)^3}{x^3}-3 i a^3 \left (-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )+\text {arccosh}(a x)^2 \log \left (1-i e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(a x)}\right )\right )\right ) \]

input

Integrate[ArcCosh[a*x]^3/x^4,x]
 

output

((6*a^2*ArcCosh[a*x])/x + (3*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCos 
h[a*x]^2)/x^2 - (2*ArcCosh[a*x]^3)/x^3 - (3*I)*a^3*((-4*I)*ArcTan[Tanh[Arc 
Cosh[a*x]/2]] + ArcCosh[a*x]^2*Log[1 - I/E^ArcCosh[a*x]] - ArcCosh[a*x]^2* 
Log[1 + I/E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2, (-I)/E^ArcCosh[a*x]] 
 - 2*ArcCosh[a*x]*PolyLog[2, I/E^ArcCosh[a*x]] + 2*PolyLog[3, (-I)/E^ArcCo 
sh[a*x]] - 2*PolyLog[3, I/E^ArcCosh[a*x]]))/6
 

3.1.30.3 Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6298, 6348, 6298, 103, 218, 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.

\(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6348

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcCosh[a*x]^3/x^4,x]
 

output

-1/3*ArcCosh[a*x]^3/x^3 + a*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2) 
/(2*x^2) - a*(-(ArcCosh[a*x]/x) + a*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]]) 
+ (a^2*(2*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] + (2*I)*(-(ArcCosh[a*x]*Po 
lyLog[2, (-I)*E^ArcCosh[a*x]]) + PolyLog[3, (-I)*E^ArcCosh[a*x]]) - (2*I)* 
(-(ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]]) + PolyLog[3, I*E^ArcCosh[a*x 
]])))/2)
 

3.1.30.3.1 Defintions of rubi rules used

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 6348

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) 
*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( 
m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)* 
(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f 
*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] 
   Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos 
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq 
Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[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]
 

3.1.30.4 Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{4}}d x\]

input

int(arccosh(a*x)^3/x^4,x)
 

output

int(arccosh(a*x)^3/x^4,x)
 

3.1.30.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arccosh(a*x)^3/x^4,x, algorithm="fricas")
 

output

integral(arccosh(a*x)^3/x^4, x)
 

3.1.30.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{4}}\, dx \]

input

integrate(acosh(a*x)**3/x**4,x)
 

output

Integral(acosh(a*x)**3/x**4, x)
 

3.1.30.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arccosh(a*x)^3/x^4,x, algorithm="maxima")
 

output

-1/3*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^3 + integrate((a^3*x^2 + s 
qrt(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^6 - a*x^4 + (a^2*x^5 - x^3)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
 

3.1.30.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arccosh(a*x)^3/x^4,x, algorithm="giac")
 

output

integrate(arccosh(a*x)^3/x^4, x)
 

3.1.30.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^4} \,d x \]

input

int(acosh(a*x)^3/x^4,x)
 

output

int(acosh(a*x)^3/x^4, x)