\(\int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [228]

Optimal result

Integrand size = 24, antiderivative size = 167 \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=-\frac {a \sqrt {1-a x} \text {arccosh}(a x)^3}{\sqrt {-1+a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{x}+\frac {3 a \sqrt {1-a x} \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {3 a \sqrt {1-a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}-\frac {3 a \sqrt {1-a x} \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )}{2 \sqrt {-1+a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.82 \[ \int \frac {\text {arccosh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (2 \text {arccosh}(a x)^2 \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )\right )}{2 \sqrt {-((-1+a x) (1+a x))}} \] Input:

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

Output:

(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(2*ArcCosh[a*x]^2*(-ArcCosh[a*x] + 
 (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x])/(a*x) - 3*Log[1 + E^( 
-2*ArcCosh[a*x])]) + 6*ArcCosh[a*x]*PolyLog[2, -E^(-2*ArcCosh[a*x])] + 3*P 
olyLog[3, -E^(-2*ArcCosh[a*x])]))/(2*Sqrt[-((-1 + a*x)*(1 + a*x))])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6332, 6297, 3042, 26, 4201, 2620, 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.

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

\(\Big \downarrow \) 6332

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 6332

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[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, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3 
, 0] && NeQ[m, -1]
 

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]
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.87

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \operatorname {arccosh}\left (a x \right )^{3}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{3} a}{a^{2} x^{2}-1}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{2 \left (a^{2} x^{2}-1\right )}\)	\(313\)

Input:

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

Output:

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

Fricas [F]

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

integrate(arccosh(a*x)^3/x^2/(-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^4 - x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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