3.1.29 \(\int \frac {\text {arccosh}(a x)^3}{x^3} \, dx\) [29]

3.1.29.1 Optimal result

Integrand size = 10, antiderivative size = 98 \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=\frac {3}{2} a^2 \text {arccosh}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-3 a^2 \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right ) \]

output

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

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

Time = 0.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=\frac {1}{2} \left (-\frac {\text {arccosh}(a x)^3}{x^2}+3 a^2 \left (\text {arccosh}(a x) \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )\right )\right ) \]

input

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

output

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

3.1.29.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6298, 6333, 6297, 3042, 26, 4201, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6333

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.1.29.3.1 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 2715

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

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 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 6333

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[b*c*(n/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Sim 
p[(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*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, 
 e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] 
 && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
 

3.1.29.4 Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2} \left (-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +3 a^{2} x^{2}+\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+3 \operatorname {arccosh}\left (a x \right )^{2}-3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\)	\(116\)
default	\(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2} \left (-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +3 a^{2} x^{2}+\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+3 \operatorname {arccosh}\left (a x \right )^{2}-3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\)	\(116\)

input

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

output

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

3.1.29.5 Fricas [F]

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

input

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

output

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

3.1.29.6 Sympy [F]

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

input

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

output

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

3.1.29.7 Maxima [F]

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

input

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

output

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

3.1.29.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.1.29.9 Mupad [F(-1)]

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

input

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

output

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