\(\int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx\) [348]

Optimal result

Integrand size = 23, antiderivative size = 210 \[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=-\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \] Output:

-3/2*a*arcsinh(a*x)^2/x-1/2*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^3/x^2-6*a^2*arc 
sinh(a*x)*arctanh(a*x+(a^2*x^2+1)^(1/2))+a^2*arcsinh(a*x)^3*arctanh(a*x+(a 
^2*x^2+1)^(1/2))-3*a^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))+3/2*a^2*arcsinh(a 
*x)^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))+3*a^2*polylog(2,a*x+(a^2*x^2+1)^(1 
/2))-3/2*a^2*arcsinh(a*x)^2*polylog(2,a*x+(a^2*x^2+1)^(1/2))-3*a^2*arcsinh 
(a*x)*polylog(3,-a*x-(a^2*x^2+1)^(1/2))+3*a^2*arcsinh(a*x)*polylog(3,a*x+( 
a^2*x^2+1)^(1/2))+3*a^2*polylog(4,-a*x-(a^2*x^2+1)^(1/2))-3*a^2*polylog(4, 
a*x+(a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 3.00 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.45 \[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\frac {a \left (-a \pi ^4 x+2 a x \text {arcsinh}(a x)^4-12 a x \text {arcsinh}(a x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-2 a x \text {arcsinh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )+48 a x \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )-48 a x \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )+8 a x \text {arcsinh}(a x)^3 \log \left (1+e^{-\text {arcsinh}(a x)}\right )-8 a x \text {arcsinh}(a x)^3 \log \left (1-e^{\text {arcsinh}(a x)}\right )-24 a x \left (-2+\text {arcsinh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-48 a x \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-24 a x \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-48 a x \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )+48 a x \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-48 a x \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(a x)}\right )-48 a x \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right )+12 a x \text {arcsinh}(a x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x)^3 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right )}{16 x} \] Input:

Integrate[ArcSinh[a*x]^3/(x^3*Sqrt[1 + a^2*x^2]),x]
 

Output:

(a*(-(a*Pi^4*x) + 2*a*x*ArcSinh[a*x]^4 - 12*a*x*ArcSinh[a*x]^2*Coth[ArcSin 
h[a*x]/2] - 2*a*x*ArcSinh[a*x]^3*Csch[ArcSinh[a*x]/2]^2 + 48*a*x*ArcSinh[a 
*x]*Log[1 - E^(-ArcSinh[a*x])] - 48*a*x*ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a 
*x])] + 8*a*x*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] - 8*a*x*ArcSinh[a* 
x]^3*Log[1 - E^ArcSinh[a*x]] - 24*a*x*(-2 + ArcSinh[a*x]^2)*PolyLog[2, -E^ 
(-ArcSinh[a*x])] - 48*a*x*PolyLog[2, E^(-ArcSinh[a*x])] - 24*a*x*ArcSinh[a 
*x]^2*PolyLog[2, E^ArcSinh[a*x]] - 48*a*x*ArcSinh[a*x]*PolyLog[3, -E^(-Arc 
Sinh[a*x])] + 48*a*x*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - 48*a*x*Poly 
Log[4, -E^(-ArcSinh[a*x])] - 48*a*x*PolyLog[4, E^ArcSinh[a*x]] + 12*a*x*Ar 
cSinh[a*x]^2*Tanh[ArcSinh[a*x]/2] - 4*ArcSinh[a*x]^3*Tanh[ArcSinh[a*x]/2]) 
)/(16*x)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.65 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6224, 6191, 6231, 3042, 26, 4670, 2715, 2838, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[ArcSinh[a*x]^3/(x^3*Sqrt[1 + a^2*x^2]),x]
 

Output:

-1/2*(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/x^2 + (3*a*(-(ArcSinh[a*x]^2/x) + 
(2*I)*a*((2*I)*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + I*PolyLog[2, -E^ArcS 
inh[a*x]] - I*PolyLog[2, E^ArcSinh[a*x]])))/2 - (I/2)*a^2*((2*I)*ArcSinh[a 
*x]^3*ArcTanh[E^ArcSinh[a*x]] - (3*I)*(-(ArcSinh[a*x]^2*PolyLog[2, -E^ArcS 
inh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] - PolyLog[4, -E^A 
rcSinh[a*x]])) + (3*I)*(-(ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]]) + 2*( 
ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - PolyLog[4, E^ArcSinh[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 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 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 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 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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6191

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

rule 6224

Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Sim 
p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 
1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 6231

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

Time = 1.69 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.80

method	result	size
default	\(-\frac {\operatorname {arcsinh}\left (x a \right )^{2} \left (\operatorname {arcsinh}\left (x a \right ) x^{2} a^{2}+3 x a \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (x a \right )\right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}+\frac {a^{2} \operatorname {arcsinh}\left (x a \right )^{3} \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {3 a^{2} \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )}{2}-3 a^{2} \operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (3, -x a -\sqrt {a^{2} x^{2}+1}\right )+3 a^{2} \operatorname {polylog}\left (4, -x a -\sqrt {a^{2} x^{2}+1}\right )-\frac {a^{2} \operatorname {arcsinh}\left (x a \right )^{3} \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )}{2}-\frac {3 a^{2} \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )}{2}+3 a^{2} \operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (3, x a +\sqrt {a^{2} x^{2}+1}\right )-3 a^{2} \operatorname {polylog}\left (4, x a +\sqrt {a^{2} x^{2}+1}\right )-3 a^{2} \operatorname {arcsinh}\left (x a \right ) \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )-3 a^{2} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )+3 a^{2} \operatorname {arcsinh}\left (x a \right ) \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+3 a^{2} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )\)	\(377\)

Input:

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

Output:

-1/2/(a^2*x^2+1)^(1/2)/x^2*arcsinh(x*a)^2*(arcsinh(x*a)*x^2*a^2+3*x*a*(a^2 
*x^2+1)^(1/2)+arcsinh(x*a))+1/2*a^2*arcsinh(x*a)^3*ln(1+x*a+(a^2*x^2+1)^(1 
/2))+3/2*a^2*arcsinh(x*a)^2*polylog(2,-x*a-(a^2*x^2+1)^(1/2))-3*a^2*arcsin 
h(x*a)*polylog(3,-x*a-(a^2*x^2+1)^(1/2))+3*a^2*polylog(4,-x*a-(a^2*x^2+1)^ 
(1/2))-1/2*a^2*arcsinh(x*a)^3*ln(1-x*a-(a^2*x^2+1)^(1/2))-3/2*a^2*arcsinh( 
x*a)^2*polylog(2,x*a+(a^2*x^2+1)^(1/2))+3*a^2*arcsinh(x*a)*polylog(3,x*a+( 
a^2*x^2+1)^(1/2))-3*a^2*polylog(4,x*a+(a^2*x^2+1)^(1/2))-3*a^2*arcsinh(x*a 
)*ln(1+x*a+(a^2*x^2+1)^(1/2))-3*a^2*polylog(2,-x*a-(a^2*x^2+1)^(1/2))+3*a^ 
2*arcsinh(x*a)*ln(1-x*a-(a^2*x^2+1)^(1/2))+3*a^2*polylog(2,x*a+(a^2*x^2+1) 
^(1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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