\(\int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [136]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {a \sqrt {1-a x}}{2 x \sqrt {-1+a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 x^2}-\frac {a^2 \sqrt {1-a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {-1+a x}}+\frac {i a^2 \sqrt {1-a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-1+a x}}-\frac {i a^2 \sqrt {1-a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {-1+a x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.39 \[ \int \frac {\text {arccosh}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {(1+a x) \left (a x \sqrt {\frac {-1+a x}{1+a x}}-\text {arccosh}(a x)+a x \text {arccosh}(a x)-i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+i a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )}{2 x^2 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6347, 15, 6361, 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.

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

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.08

method	result	size
default	\(-\frac {\left (a^{2} x^{2} \operatorname {arccosh}\left (a x \right )+\sqrt {a x -1}\, \sqrt {a x +1}\, a x -\operatorname {arccosh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \left (a^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}-\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) a^{2}}{2 a^{2} x^{2}-2}\)	\(349\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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