3.1.17 \(\int \frac {\text {arcsinh}(a x)^2}{x} \, dx\) [17]

3.1.17.1 Optimal result

Integrand size = 10, antiderivative size = 60 \[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=-\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \]

output

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

3.1.17.2 Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=-\frac {1}{3} \text {arcsinh}(a x)^3+\text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+\text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \]

input

Integrate[ArcSinh[a*x]^2/x,x]
 

output

-1/3*ArcSinh[a*x]^3 + ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + ArcSinh 
[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - PolyLog[3, E^(2*ArcSinh[a*x])]/2
 

3.1.17.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.28, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6190, 3042, 26, 4199, 25, 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.

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSinh[a*x]^2/x,x]
 

output

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

3.1.17.3.1 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]
 

3.1.17.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.52

method	result	size
derivativedivides	\(-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{3}+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\)	\(151\)
default	\(-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{3}+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\)	\(151\)

input

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

output

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

3.1.17.5 Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x} \,d x } \]

input

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

output

integral(arcsinh(a*x)^2/x, x)
 

3.1.17.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x}\, dx \]

input

integrate(asinh(a*x)**2/x,x)
 

output

Integral(asinh(a*x)**2/x, x)
 

3.1.17.7 Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x} \,d x } \]

input

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

output

integrate(arcsinh(a*x)^2/x, x)
 

3.1.17.8 Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x} \,d x } \]

input

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

output

integrate(arcsinh(a*x)^2/x, x)
 

3.1.17.9 Mupad [F(-1)]

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

input

int(asinh(a*x)^2/x,x)
 

output

int(asinh(a*x)^2/x, x)