\(\int \frac {\text {arcsinh}(a x)^4}{x} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 97 \[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=-\frac {1}{5} \text {arcsinh}(a x)^5+\text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+2 \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (5,e^{2 \text {arcsinh}(a x)}\right ) \] Output:

-1/5*arcsinh(a*x)^5+arcsinh(a*x)^4*ln(1-(a*x+(a^2*x^2+1)^(1/2))^2)+2*arcsi 
nh(a*x)^3*polylog(2,(a*x+(a^2*x^2+1)^(1/2))^2)-3*arcsinh(a*x)^2*polylog(3, 
(a*x+(a^2*x^2+1)^(1/2))^2)+3*arcsinh(a*x)*polylog(4,(a*x+(a^2*x^2+1)^(1/2) 
)^2)-3/2*polylog(5,(a*x+(a^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=-\frac {1}{5} \text {arcsinh}(a x)^5+\text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+2 \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )+3 \text {arcsinh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (5,e^{2 \text {arcsinh}(a x)}\right ) \] Input:

Integrate[ArcSinh[a*x]^4/x,x]
 

Output:

-1/5*ArcSinh[a*x]^5 + ArcSinh[a*x]^4*Log[1 - E^(2*ArcSinh[a*x])] + 2*ArcSi 
nh[a*x]^3*PolyLog[2, E^(2*ArcSinh[a*x])] - 3*ArcSinh[a*x]^2*PolyLog[3, E^( 
2*ArcSinh[a*x])] + 3*ArcSinh[a*x]*PolyLog[4, E^(2*ArcSinh[a*x])] - (3*Poly 
Log[5, E^(2*ArcSinh[a*x])])/2
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.29, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6190, 3042, 26, 4199, 25, 2620, 3011, 7163, 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.

\(\displaystyle \int \frac {\text {arcsinh}(a x)^4}{x} \, dx\)

\(\Big \downarrow \) 6190

\(\displaystyle \int \frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^4}{a x}d\text {arcsinh}(a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -i \text {arcsinh}(a x)^4 \tan \left (\frac {\pi }{2}+i \text {arcsinh}(a x)\right )d\text {arcsinh}(a x)\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \text {arcsinh}(a x)^4 \tan \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )d\text {arcsinh}(a x)\)

\(\Big \downarrow \) 4199

\(\displaystyle -i \left (2 i \int -\frac {e^{2 \text {arcsinh}(a x)} \text {arcsinh}(a x)^4}{1-e^{2 \text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -i \left (-2 i \int \frac {e^{2 \text {arcsinh}(a x)} \text {arcsinh}(a x)^4}{1-e^{2 \text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -i \left (-2 i \left (2 \int \text {arcsinh}(a x)^3 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-\frac {1}{2} \text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -i \left (-2 i \left (2 \left (\frac {3}{2} \int \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-\frac {1}{2} \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle -i \left (-2 i \left (2 \left (\frac {3}{2} \left (\frac {1}{2} \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )-\int \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)\right )-\frac {1}{2} \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle -i \left (-2 i \left (2 \left (\frac {3}{2} \left (\frac {1}{2} \int \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+\frac {1}{2} \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -i \left (-2 i \left (2 \left (\frac {3}{2} \left (\frac {1}{4} \int e^{-2 \text {arcsinh}(a x)} \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )de^{2 \text {arcsinh}(a x)}+\frac {1}{2} \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -i \left (-2 i \left (2 \left (\frac {3}{2} \left (\frac {1}{2} \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \text {arcsinh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(a x)}\right )+\frac {1}{4} \operatorname {PolyLog}\left (5,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{2} \text {arcsinh}(a x)^4 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )\right )-\frac {1}{5} i \text {arcsinh}(a x)^5\right )\)

Input:

Int[ArcSinh[a*x]^4/x,x]
 

Output:

(-I)*((-1/5*I)*ArcSinh[a*x]^5 - (2*I)*(-1/2*(ArcSinh[a*x]^4*Log[1 - E^(2*A 
rcSinh[a*x])]) + 2*(-1/2*(ArcSinh[a*x]^3*PolyLog[2, E^(2*ArcSinh[a*x])]) + 
 (3*((ArcSinh[a*x]^2*PolyLog[3, E^(2*ArcSinh[a*x])])/2 - (ArcSinh[a*x]*Pol 
yLog[4, E^(2*ArcSinh[a*x])])/2 + PolyLog[5, E^(2*ArcSinh[a*x])]/4))/2)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

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 = 0.70 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.65

method	result	size
derivativedivides	\(-\frac {\operatorname {arcsinh}\left (x a \right )^{5}}{5}+\operatorname {arcsinh}\left (x a \right )^{4} \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {arcsinh}\left (x a \right )^{3} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (3, -x a -\sqrt {a^{2} x^{2}+1}\right )+24 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (4, -x a -\sqrt {a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, -x a -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x a \right )^{4} \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {arcsinh}\left (x a \right )^{3} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (3, x a +\sqrt {a^{2} x^{2}+1}\right )+24 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (4, x a +\sqrt {a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, x a +\sqrt {a^{2} x^{2}+1}\right )\)	\(257\)
default	\(-\frac {\operatorname {arcsinh}\left (x a \right )^{5}}{5}+\operatorname {arcsinh}\left (x a \right )^{4} \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {arcsinh}\left (x a \right )^{3} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (3, -x a -\sqrt {a^{2} x^{2}+1}\right )+24 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (4, -x a -\sqrt {a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, -x a -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x a \right )^{4} \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+4 \operatorname {arcsinh}\left (x a \right )^{3} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {arcsinh}\left (x a \right )^{2} \operatorname {polylog}\left (3, x a +\sqrt {a^{2} x^{2}+1}\right )+24 \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (4, x a +\sqrt {a^{2} x^{2}+1}\right )-24 \operatorname {polylog}\left (5, x a +\sqrt {a^{2} x^{2}+1}\right )\)	\(257\)

Input:

int(arcsinh(x*a)^4/x,x,method=_RETURNVERBOSE)
 

Output:

-1/5*arcsinh(x*a)^5+arcsinh(x*a)^4*ln(1+x*a+(a^2*x^2+1)^(1/2))+4*arcsinh(x 
*a)^3*polylog(2,-x*a-(a^2*x^2+1)^(1/2))-12*arcsinh(x*a)^2*polylog(3,-x*a-( 
a^2*x^2+1)^(1/2))+24*arcsinh(x*a)*polylog(4,-x*a-(a^2*x^2+1)^(1/2))-24*pol 
ylog(5,-x*a-(a^2*x^2+1)^(1/2))+arcsinh(x*a)^4*ln(1-x*a-(a^2*x^2+1)^(1/2))+ 
4*arcsinh(x*a)^3*polylog(2,x*a+(a^2*x^2+1)^(1/2))-12*arcsinh(x*a)^2*polylo 
g(3,x*a+(a^2*x^2+1)^(1/2))+24*arcsinh(x*a)*polylog(4,x*a+(a^2*x^2+1)^(1/2) 
)-24*polylog(5,x*a+(a^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x} \,d x } \] Input:

integrate(arcsinh(a*x)^4/x,x, algorithm="fricas")
 

Output:

integral(arcsinh(a*x)^4/x, x)
 

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=\int \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{x}\, dx \] Input:

integrate(asinh(a*x)**4/x,x)
 

Output:

Integral(asinh(a*x)**4/x, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x} \,d x } \] Input:

integrate(arcsinh(a*x)^4/x,x, algorithm="maxima")
 

Output:

integrate(arcsinh(a*x)^4/x, x)
 

Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x} \,d x } \] Input:

integrate(arcsinh(a*x)^4/x,x, algorithm="giac")
 

Output:

integrate(arcsinh(a*x)^4/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{x} \,d x \] Input:

int(asinh(a*x)^4/x,x)
 

Output:

int(asinh(a*x)^4/x, x)
 

Reduce [F]

\[ \int \frac {\text {arcsinh}(a x)^4}{x} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{4}}{x}d x \] Input:

int(asinh(a*x)^4/x,x)
 

Output:

int(asinh(a*x)**4/x,x)