\(\int \frac {\text {arcsinh}(a x)^2}{x^2 \sqrt {1+a^2 x^2}} \, dx\) [294]

Optimal result

Integrand size = 23, antiderivative size = 66 \[ \int \frac {\text {arcsinh}(a x)^2}{x^2 \sqrt {1+a^2 x^2}} \, dx=-a \text {arcsinh}(a x)^2-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{x}+2 a \text {arcsinh}(a x) \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+a \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {\text {arcsinh}(a x)^2}{x^2 \sqrt {1+a^2 x^2}} \, dx=a \left (\text {arcsinh}(a x) \left (\text {arcsinh}(a x)-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a x}+2 \log \left (1-e^{-2 \text {arcsinh}(a x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(a x)}\right )\right ) \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {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.

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.00

method	result	size
default	\(\frac {\left (x a -\sqrt {a^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x a \right )^{2}}{x}-2 a \operatorname {arcsinh}\left (x a \right )^{2}+2 a \,\operatorname {arcsinh}\left (x a \right ) \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )+2 a \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )+2 a \,\operatorname {arcsinh}\left (x a \right ) \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )+2 a \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )\)	\(132\)

Input:

int(arcsinh(x*a)^2/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)^2-2*a*arcsinh(x*a)^2+2*a*arcsinh(x* 
a)*ln(1-x*a-(a^2*x^2+1)^(1/2))+2*a*polylog(2,x*a+(a^2*x^2+1)^(1/2))+2*a*ar 
csinh(x*a)*ln(1+x*a+(a^2*x^2+1)^(1/2))+2*a*polylog(2,-x*a-(a^2*x^2+1)^(1/2 
))
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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