3.3.32 \(\int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [232]

3.3.32.1 Optimal result

Integrand size = 24, antiderivative size = 296 \[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {-1+a x} \text {arccosh}(a x)}{x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {a^2 \sqrt {-1+a x} \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \]

output

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

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

Time = 0.78 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.79 \[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {i a^2 \sqrt {-((-1+a x) (1+a x))} \left (\frac {2 i \text {arccosh}(a x)}{a x}+\frac {i \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^2}{a^2 x^2}-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )+\text {arccosh}(a x)^2 \log \left (1-i e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(a x)}\right )\right )}{2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]

input

Integrate[ArcCosh[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]
 

output

((I/2)*a^2*Sqrt[-((-1 + a*x)*(1 + a*x))]*(((2*I)*ArcCosh[a*x])/(a*x) + (I* 
Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^2)/(a^2*x^2) - (4*I)*Arc 
Tan[Tanh[ArcCosh[a*x]/2]] + ArcCosh[a*x]^2*Log[1 - I/E^ArcCosh[a*x]] - Arc 
Cosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2, (-I)/E^A 
rcCosh[a*x]] - 2*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]]))/(Sqrt[(-1 + a*x)/( 
1 + a*x)]*(1 + a*x))
 

3.3.32.3 Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.66, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6347, 6298, 103, 218, 6361, 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.

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

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcCosh[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]
 

output

-1/2*(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/x^2 - (a*Sqrt[-1 + a*x]*(-(ArcCosh 
[a*x]/x) + a*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]]))/Sqrt[1 - a*x] + (a^2*S 
qrt[-1 + a*x]*(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^ArcCosh[a*x]]) - 
 (2*I)*(-(ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]]) + PolyLog[3, I*E^ArcC 
osh[a*x]])))/(2*Sqrt[1 - a*x])
 

3.3.32.3.1 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 6347

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 
)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp 
[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

rule 6361

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x 
]/Sqrt[d + e*x^2])]   Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] 
, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int 
egerQ[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]
 

3.3.32.4 Maple [F]

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

input

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

output

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

3.3.32.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]

input

integrate(arccosh(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

output

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^2/(a^2*x^5 - x^3), x)
 

3.3.32.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

input

integrate(acosh(a*x)**2/x**3/(-a**2*x**2+1)**(1/2),x)
 

output

Integral(acosh(a*x)**2/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

3.3.32.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]

input

integrate(arccosh(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

output

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

3.3.32.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]

input

integrate(arccosh(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

output

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

3.3.32.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]

input

int(acosh(a*x)^2/(x^3*(1 - a^2*x^2)^(1/2)),x)
 

output

int(acosh(a*x)^2/(x^3*(1 - a^2*x^2)^(1/2)), x)