\(\int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx\) [128]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \] Output:

-1/2*a/x-1/2*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/x^2+a^2*arcsinh(a*x)*arctanh(a 
*x+(a^2*x^2+1)^(1/2))+1/2*a^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))-1/2*a^2*po 
lylog(2,a*x+(a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\frac {1}{8} a^2 \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-\text {arcsinh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )+4 \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right ) \] Input:

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

Output:

(a^2*(-2*Coth[ArcSinh[a*x]/2] - ArcSinh[a*x]*Csch[ArcSinh[a*x]/2]^2 - 4*Ar 
cSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] + 4*ArcSinh[a*x]*Log[1 + E^(-ArcSinh 
[a*x])] - 4*PolyLog[2, -E^(-ArcSinh[a*x])] + 4*PolyLog[2, E^(-ArcSinh[a*x] 
)] - ArcSinh[a*x]*Sech[ArcSinh[a*x]/2]^2 + 2*Tanh[ArcSinh[a*x]/2]))/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6224, 15, 6231, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/2*a/x - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*x^2) - (I/2)*a^2*((2*I)*Arc 
Sinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + I*PolyLog[2, -E^ArcSinh[a*x]] - I*Poly 
Log[2, E^ArcSinh[a*x]])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

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]
 

rule 6231

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e 
*x^2]]   Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ 
[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.88

method	result	size
default	\(-\frac {\operatorname {arcsinh}\left (x a \right ) x^{2} a^{2}+x a \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (x a \right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}-\frac {a^{2} \operatorname {arcsinh}\left (x a \right ) \ln \left (1-x a -\sqrt {a^{2} x^{2}+1}\right )}{2}-\frac {a^{2} \operatorname {polylog}\left (2, x a +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \operatorname {arcsinh}\left (x a \right ) \ln \left (1+x a +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \operatorname {polylog}\left (2, -x a -\sqrt {a^{2} x^{2}+1}\right )}{2}\)	\(150\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(a^2*x^2 + 1)*arcsinh(a*x)/(a^2*x^5 + x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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