\(\int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx\) [296]

Optimal result

Integrand size = 23, antiderivative size = 145 \[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=-\frac {a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {a \text {arcsinh}(a x)}{3 x^2}+\frac {2}{3} a^3 \text {arcsinh}(a x)^2-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 x^3}+\frac {2 a^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{3 x}-\frac {4}{3} a^3 \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\frac {2}{3} a^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\frac {1}{3} a^3 \left (-\frac {\sqrt {1+a^2 x^2} \left (1+\left (-2+\frac {1}{a^2 x^2}\right ) \text {arcsinh}(a x)^2\right )}{a x}-\text {arcsinh}(a x) \left (\frac {1}{a^2 x^2}+2 \text {arcsinh}(a x)+4 \log \left (1-e^{-2 \text {arcsinh}(a x)}\right )\right )+2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(a x)}\right )\right ) \] Input:

Integrate[ArcSinh[a*x]^2/(x^4*Sqrt[1 + a^2*x^2]),x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6224, 6191, 242, 6215, 6190, 3042, 26, 4199, 25, 2620, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcSinh[a*x]^2/(x^4*Sqrt[1 + a^2*x^2]),x]
 

Output:

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

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 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -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 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 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 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 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 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]
 

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.94

method	result	size
default	\(-\frac {\left (2 x^{3} a^{3}-2 x^{2} a^{2} \sqrt {a^{2} x^{2}+1}+\sqrt {a^{2} x^{2}+1}\right ) \left (2 x^{4} a^{4} \operatorname {arcsinh}\left (x a \right )+2 x^{3} a^{3} \operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}+a^{4} x^{4}+x^{3} a^{3} \sqrt {a^{2} x^{2}+1}+3 a^{2} x^{2} \operatorname {arcsinh}\left (x a \right )^{2}-\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a -a^{2} x^{2}-\operatorname {arcsinh}\left (x a \right )^{2}\right )}{3 \left (3 a^{2} x^{2}-1\right ) x^{3}}+\frac {4 a^{3} \operatorname {arcsinh}\left (x a \right )^{2}}{3}-\frac {4 a^{3} \operatorname {arcsinh}\left (x a \right ) \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {4 a^{3} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {4 a^{3} \operatorname {arcsinh}\left (x a \right ) \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )}{3}-\frac {4 a^{3} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )}{3}\)	\(281\)

Input:

int(arcsinh(x*a)^2/x^4/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x^{4}} \,d x } \] Input:

integrate(arcsinh(a*x)^2/x^4/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(a^2*x^2 + 1)*arcsinh(a*x)^2/(a^2*x^6 + x^4), x)
 

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{4} \sqrt {a^{2} x^{2} + 1}}\, dx \] Input:

integrate(asinh(a*x)**2/x**4/(a**2*x**2+1)**(1/2),x)
 

Output:

Integral(asinh(a*x)**2/(x**4*sqrt(a**2*x**2 + 1)), x)
 

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} x^{4}} \,d x } \] Input:

integrate(arcsinh(a*x)^2/x^4/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

1/3*(2*a^4*x^4 + a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 + 1))^2/(sqrt(a^2*x^2 
 + 1)*x^3) - integrate(2/3*(2*a^5*x^4 + a^3*x^2 + (2*a^4*x^3 - a^2*x)*sqrt 
(a^2*x^2 + 1) - a)*log(a*x + sqrt(a^2*x^2 + 1))/(sqrt(a^2*x^2 + 1)*a*x^4 + 
 (a^2*x^2 + 1)*x^3), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arcsinh(a*x)^2/x^4/(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. \[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^4\,\sqrt {a^2\,x^2+1}} \,d x \] Input:

int(asinh(a*x)^2/(x^4*(a^2*x^2 + 1)^(1/2)),x)
 

Output:

int(asinh(a*x)^2/(x^4*(a^2*x^2 + 1)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\text {arcsinh}(a x)^2}{x^4 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{2}}{\sqrt {a^{2} x^{2}+1}\, x^{4}}d x \] Input:

int(asinh(a*x)^2/x^4/(a^2*x^2+1)^(1/2),x)
 

Output:

int(asinh(a*x)**2/(sqrt(a**2*x**2 + 1)*x**4),x)