\(\int \frac {\text {arccosh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\) [227]

Optimal result

Integrand size = 24, antiderivative size = 265 \[ \int \frac {\text {arccosh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {1-a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {3 i \sqrt {1-a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {3 i \sqrt {1-a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {6 i \sqrt {1-a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {6 i \sqrt {1-a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {6 i \sqrt {1-a x} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {6 i \sqrt {1-a x} \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}} \] Output:

-2*(-a*x+1)^(1/2)*arccosh(a*x)^3*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/( 
a*x-1)^(1/2)+3*I*(-a*x+1)^(1/2)*arccosh(a*x)^2*polylog(2,-I*(a*x+(a*x-1)^( 
1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)-3*I*(-a*x+1)^(1/2)*arccosh(a*x)^2*polyl 
og(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)-6*I*(-a*x+1)^(1/2) 
*arccosh(a*x)*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2 
)+6*I*(-a*x+1)^(1/2)*arccosh(a*x)*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^( 
1/2)))/(a*x-1)^(1/2)+6*I*(-a*x+1)^(1/2)*polylog(4,-I*(a*x+(a*x-1)^(1/2)*(a 
*x+1)^(1/2)))/(a*x-1)^(1/2)-6*I*(-a*x+1)^(1/2)*polylog(4,I*(a*x+(a*x-1)^(1 
/2)*(a*x+1)^(1/2)))/(a*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.84 \[ \int \frac {\text {arccosh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\frac {i \sqrt {-((-1+a x) (1+a x))} \left (7 \pi ^4+8 i \pi ^3 \text {arccosh}(a x)+24 \pi ^2 \text {arccosh}(a x)^2-32 i \pi \text {arccosh}(a x)^3-16 \text {arccosh}(a x)^4+8 i \pi ^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )+48 \pi ^2 \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-96 i \pi \text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-64 \text {arccosh}(a x)^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-48 \pi ^2 \text {arccosh}(a x) \log \left (1-i e^{\text {arccosh}(a x)}\right )+96 i \pi \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+64 \text {arccosh}(a x)^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arccosh}(a x))\right )\right )-48 (\pi -2 i \text {arccosh}(a x))^2 \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+192 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-48 \pi ^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+384 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-384 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-192 i \pi \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{-\text {arccosh}(a x)}\right )+384 \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )}{64 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:

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

Output:

((I/64)*Sqrt[-((-1 + a*x)*(1 + a*x))]*(7*Pi^4 + (8*I)*Pi^3*ArcCosh[a*x] + 
24*Pi^2*ArcCosh[a*x]^2 - (32*I)*Pi*ArcCosh[a*x]^3 - 16*ArcCosh[a*x]^4 + (8 
*I)*Pi^3*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*ArcCosh[a*x]*Log[1 + I/E^ArcC 
osh[a*x]] - (96*I)*Pi*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*ArcCos 
h[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*ArcCosh[a*x]*Log[1 - I*E^ArcC 
osh[a*x]] + (96*I)*Pi*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I)*Pi^ 
3*Log[1 + I*E^ArcCosh[a*x]] + 64*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] 
+ (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4]] - 48*(Pi - (2*I)*ArcCos 
h[a*x])^2*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 192*ArcCosh[a*x]^2*PolyLog[2, 
(-I)*E^ArcCosh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*A 
rcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^Ar 
cCosh[a*x]] + 384*ArcCosh[a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 384*ArcCo 
sh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcCos 
h[a*x]] + 384*PolyLog[4, (-I)/E^ArcCosh[a*x]] + 384*PolyLog[4, (-I)*E^ArcC 
osh[a*x]]))/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.58, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6361, 3042, 4668, 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.

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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