3.1.73 \(\int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx\) [73]

3.1.73.1 Optimal result

Integrand size = 12, antiderivative size = 235 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=-\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{2 x^2}+\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1+a^2}+\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}}-\frac {a b^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{3/2}} \]

output

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

3.1.73.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.19 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^3} \, dx=-\frac {2 \sqrt {1+a^2} b x \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)+\sqrt {1+a^2} \text {arcsinh}(a+b x)^2+a^2 \sqrt {1+a^2} \text {arcsinh}(a+b x)^2+2 a b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-2 a b^2 x^2 \text {arcsinh}(a+b x) \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )-2 \sqrt {1+a^2} b^2 x^2 \log (x)-2 a b^2 x^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 a b^2 x^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{2 \left (1+a^2\right )^{3/2} x^2} \]

input

Integrate[ArcSinh[a + b*x]^2/x^3,x]
 

output

-1/2*(2*Sqrt[1 + a^2]*b*x*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x] + Sqrt[1 
+ a^2]*ArcSinh[a + b*x]^2 + a^2*Sqrt[1 + a^2]*ArcSinh[a + b*x]^2 + 2*a*b^2 
*x^2*ArcSinh[a + b*x]*Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sq 
rt[1 + a^2])] - 2*a*b^2*x^2*ArcSinh[a + b*x]*Log[(-a + Sqrt[1 + a^2] + E^A 
rcSinh[a + b*x])/(-a + Sqrt[1 + a^2])] - 2*Sqrt[1 + a^2]*b^2*x^2*Log[x] - 
2*a*b^2*x^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 2*a*b^2*x 
^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/((1 + a^2)^(3/2)*x^ 
2)
 

3.1.73.3 Rubi [A] (warning: unable to verify)

Time = 1.07 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6274, 25, 27, 6243, 6258, 3042, 3805, 3042, 3147, 16, 3803, 2694, 27, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3147

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[ArcSinh[a + b*x]^2/x^3,x]
 

output

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

3.1.73.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( 
-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; 
FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 6243

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

rule 6258

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S 
qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[I 
nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b 
, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt 
Q[m, 0] || IGtQ[n, 0])
 

rule 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.1.73.4 Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (-2 \left (b x +a \right )^{2}+4 a \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )+a^{2} \operatorname {arcsinh}\left (b x +a \right )-2 a \sqrt {1+\left (b x +a \right )^{2}}+2 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}-2 a^{2}\right )}{2 b^{2} x^{2} \left (a^{2}+1\right )}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {2 \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}+\frac {\ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}+1}\right )\)	\(374\)
default	\(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (-2 \left (b x +a \right )^{2}+4 a \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )+a^{2} \operatorname {arcsinh}\left (b x +a \right )-2 a \sqrt {1+\left (b x +a \right )^{2}}+2 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}-2 a^{2}\right )}{2 b^{2} x^{2} \left (a^{2}+1\right )}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}+\frac {a \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}-\frac {2 \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{a^{2}+1}+\frac {\ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}+1}\right )\)	\(374\)

input

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

output

b^2*(-1/2*arcsinh(b*x+a)*(-2*(b*x+a)^2+4*a*(b*x+a)+arcsinh(b*x+a)+a^2*arcs 
inh(b*x+a)-2*a*(1+(b*x+a)^2)^(1/2)+2*(b*x+a)*(1+(b*x+a)^2)^(1/2)-2*a^2)/b^ 
2/x^2/(a^2+1)-1/(a^2+1)^(3/2)*a*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)-b*x-(1+(b 
*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))+1/(a^2+1)^(3/2)*a*arcsinh(b*x+a)*ln(((a 
^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))-1/(a^2+1)^(3/2)*a 
*dilog(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))+1/(a^2+1 
)^(3/2)*a*dilog(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)) 
)-2/(a^2+1)*ln(b*x+a+(1+(b*x+a)^2)^(1/2))+1/(a^2+1)*ln(2*a*(b*x+a+(1+(b*x+ 
a)^2)^(1/2))-(b*x+a+(1+(b*x+a)^2)^(1/2))^2+1))
 

3.1.73.5 Fricas [F]

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

input

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

output

integral(arcsinh(b*x + a)^2/x^3, x)
 

3.1.73.6 Sympy [F]

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

input

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

output

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

3.1.73.7 Maxima [F]

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

input

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

output

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

3.1.73.8 Giac [F]

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

input

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

output

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

3.1.73.9 Mupad [F(-1)]

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

input

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

output

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