\(\int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 178 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=-\frac {\text {arcsinh}(a+b x)^2}{x}-\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^2} \, dx=\frac {-\text {arcsinh}(a+b x) \left (\sqrt {1+a^2} \text {arcsinh}(a+b x)+2 b x \left (-\log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+\log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )\right )\right )-2 b x \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+2 b x \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2} x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6274, 27, 6243, 6258, 3042, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

\(\displaystyle b \left (-4 \int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{2 e^{\text {arcsinh}(a+b x)} a-e^{2 \text {arcsinh}(a+b x)}+1}d\text {arcsinh}(a+b x)-\frac {\text {arcsinh}(a+b x)^2}{b x}\right )\)

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

b*(-(ArcSinh[a + b*x]^2/(b*x)) - 4*(-1/2*(-(ArcSinh[a + b*x]*Log[1 - E^Arc 
Sinh[a + b*x]/(a - Sqrt[1 + a^2])]) - PolyLog[2, E^ArcSinh[a + b*x]/(a - S 
qrt[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])))
 

Defintions of rubi rules used

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 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 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]
 

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(b \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{b x}+\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \left (\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{\sqrt {a^{2}+1}}+\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\right )\)	\(206\)
default	\(b \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{b x}+\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \left (\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{\sqrt {a^{2}+1}}+\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {2 \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\right )\)	\(206\)

Input:

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

Output:

b*(-arcsinh(b*x+a)^2/b/x+2*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)))-ln(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/ 
(-a+(a^2+1)^(1/2))))/(a^2+1)^(1/2)+2/(a^2+1)^(1/2)*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)^(1/2)*dilog(((a^2+1)^( 
1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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