3.1.27 \(\int \frac {\text {arccosh}(a x)^3}{x} \, dx\) [27]

3.1.27.1 Optimal result

Integrand size = 10, antiderivative size = 87 \[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=-\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {arccosh}(a x)}\right ) \]

output

-1/4*arccosh(a*x)^4+arccosh(a*x)^3*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^ 
2)+3/2*arccosh(a*x)^2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-3/2* 
arccosh(a*x)*polylog(3,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)+3/4*polylog(4 
,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)
 

3.1.27.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\frac {1}{4} \left (\text {arccosh}(a x)^4+4 \text {arccosh}(a x)^3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )-6 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )-6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )-3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arccosh}(a x)}\right )\right ) \]

input

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

output

(ArcCosh[a*x]^4 + 4*ArcCosh[a*x]^3*Log[1 + E^(-2*ArcCosh[a*x])] - 6*ArcCos 
h[a*x]^2*PolyLog[2, -E^(-2*ArcCosh[a*x])] - 6*ArcCosh[a*x]*PolyLog[3, -E^( 
-2*ArcCosh[a*x])] - 3*PolyLog[4, -E^(-2*ArcCosh[a*x])])/4
 

3.1.27.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6297, 3042, 26, 4201, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 3042

\(\displaystyle \int -i \text {arccosh}(a x)^3 \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \text {arccosh}(a x)^3 \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\)

\(\Big \downarrow \) 4201

\(\displaystyle -i \left (2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)^3}{1+e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{4} i \text {arccosh}(a x)^4\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -i \left (2 i \left (\frac {1}{2} \text {arccosh}(a x)^3 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {3}{2} \int \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\frac {1}{4} i \text {arccosh}(a x)^4\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -i \left (2 i \left (\frac {1}{2} \text {arccosh}(a x)^3 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {3}{2} \left (\int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\frac {1}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )\right )\right )-\frac {1}{4} i \text {arccosh}(a x)^4\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle -i \left (2 i \left (\frac {1}{2} \text {arccosh}(a x)^3 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {3}{2} \left (-\frac {1}{2} \int \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\frac {1}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )\right )\right )-\frac {1}{4} i \text {arccosh}(a x)^4\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -i \left (2 i \left (\frac {1}{2} \text {arccosh}(a x)^3 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {3}{2} \left (-\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}-\frac {1}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )\right )\right )-\frac {1}{4} i \text {arccosh}(a x)^4\right )\)

\(\Big \downarrow \) 7143

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

input

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

output

(-I)*((-1/4*I)*ArcCosh[a*x]^4 + (2*I)*((ArcCosh[a*x]^3*Log[1 + E^(2*ArcCos 
h[a*x])])/2 - (3*(-1/2*(ArcCosh[a*x]^2*PolyLog[2, -E^(2*ArcCosh[a*x])]) + 
(ArcCosh[a*x]*PolyLog[3, -E^(2*ArcCosh[a*x])])/2 - PolyLog[4, -E^(2*ArcCos 
h[a*x])]/4))/2))
 

3.1.27.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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

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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6297

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a 
, b, c}, 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]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.1.27.4 Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(-\frac {\operatorname {arccosh}\left (a x \right )^{4}}{4}+\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\)	\(132\)
default	\(-\frac {\operatorname {arccosh}\left (a x \right )^{4}}{4}+\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\)	\(132\)

input

int(arccosh(a*x)^3/x,x,method=_RETURNVERBOSE)
 

output

-1/4*arccosh(a*x)^4+arccosh(a*x)^3*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^ 
2)+3/2*arccosh(a*x)^2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-3/2* 
arccosh(a*x)*polylog(3,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)+3/4*polylog(4 
,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)
 

3.1.27.5 Fricas [F]

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

input

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

output

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

3.1.27.6 Sympy [F]

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

input

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

output

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

3.1.27.7 Maxima [F]

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

input

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

output

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

3.1.27.8 Giac [F]

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

input

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

output

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

3.1.27.9 Mupad [F(-1)]

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

input

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

output

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