3.1.20 \(\int \frac {\text {arccosh}(a x)^2}{x^4} \, dx\) [20]

3.1.20.1 Optimal result

Integrand size = 10, antiderivative size = 114 \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\frac {a^2}{3 x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x^2}-\frac {\text {arccosh}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right ) \]

output

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

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

Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\frac {1}{3} a^3 \left (\frac {1}{a x}+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a^2 x^2}-\frac {\text {arccosh}(a x)^2}{a^3 x^3}-i \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+i \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right ) \]

input

Integrate[ArcCosh[a*x]^2/x^4,x]
 

output

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

3.1.20.3 Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6298, 6348, 15, 6362, 3042, 4668, 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.

\(\displaystyle \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6348

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[ArcCosh[a*x]^2/x^4,x]
 

output

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

3.1.20.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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]
 

3.1.20.4 Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(a^{3} \left (-\frac {-a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+\operatorname {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}\right )\)	\(171\)
default	\(a^{3} \left (-\frac {-a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+\operatorname {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )}{3}\right )\)	\(171\)

input

int(arccosh(a*x)^2/x^4,x,method=_RETURNVERBOSE)
 

output

a^3*(-1/3*(-a*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+arccosh(a*x)^2-a^ 
2*x^2)/a^3/x^3-1/3*I*arccosh(a*x)*ln(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)) 
)+1/3*I*arccosh(a*x)*ln(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-1/3*I*dilog 
(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*I*dilog(1-I*(a*x+(a*x-1)^(1/2) 
*(a*x+1)^(1/2))))
 

3.1.20.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{4}} \,d x } \]

input

integrate(arccosh(a*x)^2/x^4,x, algorithm="fricas")
 

output

integral(arccosh(a*x)^2/x^4, x)
 

3.1.20.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{4}}\, dx \]

input

integrate(acosh(a*x)**2/x**4,x)
 

output

Integral(acosh(a*x)**2/x**4, x)
 

3.1.20.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{4}} \,d x } \]

input

integrate(arccosh(a*x)^2/x^4,x, algorithm="maxima")
 

output

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

3.1.20.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{4}} \,d x } \]

input

integrate(arccosh(a*x)^2/x^4,x, algorithm="giac")
 

output

integrate(arccosh(a*x)^2/x^4, x)
 

3.1.20.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^4} \,d x \]

input

int(acosh(a*x)^2/x^4,x)
 

output

int(acosh(a*x)^2/x^4, x)