\(\int \frac {\text {arcsinh}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx\) [347]

Optimal result

Integrand size = 23, antiderivative size = 88 \[ \int \frac {\text {arcsinh}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx=-a \text {arcsinh}(a x)^3-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}+3 a \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+3 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10 \[ \int \frac {\text {arcsinh}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx=\frac {1}{8} a \left (i \pi ^3-8 \text {arcsinh}(a x)^3-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a x}+24 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+24 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )\right ) \] Input:

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

Output:

(a*(I*Pi^3 - 8*ArcSinh[a*x]^3 - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(a*x) 
 + 24*ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + 24*ArcSinh[a*x]*PolyLog 
[2, E^(2*ArcSinh[a*x])] - 12*PolyLog[3, E^(2*ArcSinh[a*x])]))/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6215, 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.

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 6215

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[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, m, p}, x] && EqQ 
[e, c^2*d] && 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 = 1.61 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {\left (x a -\sqrt {a^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x a \right )^{3}}{x}-2 a \operatorname {arcsinh}\left (x a \right )^{3}+3 a \operatorname {arcsinh}\left (x a \right )^{2} \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )+6 a \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )-6 a \operatorname {polylog}\left (3, -x a -\sqrt {a^{2} x^{2}+1}\right )+3 a \operatorname {arcsinh}\left (x a \right )^{2} \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+6 a \,\operatorname {arcsinh}\left (x a \right ) \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )-6 a \operatorname {polylog}\left (3, x a +\sqrt {a^{2} x^{2}+1}\right )\)	\(187\)

Input:

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

Output:

(x*a-(a^2*x^2+1)^(1/2))/x*arcsinh(x*a)^3-2*a*arcsinh(x*a)^3+3*a*arcsinh(x* 
a)^2*ln(1+x*a+(a^2*x^2+1)^(1/2))+6*a*arcsinh(x*a)*polylog(2,-x*a-(a^2*x^2+ 
1)^(1/2))-6*a*polylog(3,-x*a-(a^2*x^2+1)^(1/2))+3*a*arcsinh(x*a)^2*ln(1-x* 
a-(a^2*x^2+1)^(1/2))+6*a*arcsinh(x*a)*polylog(2,x*a+(a^2*x^2+1)^(1/2))-6*a 
*polylog(3,x*a+(a^2*x^2+1)^(1/2))
 

Fricas [F]

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

integrate(arcsinh(a*x)^3/x^2/(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^4 + x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arcsinh}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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