\(\int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx\) [28]

Optimal result

Integrand size = 12, antiderivative size = 154 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {2 a \text {Chi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {\text {Shi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))}{b^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \] Output:

-a^2*(1+(b*x+a)^2)^(1/2)/b^3/arcsinh(b*x+a)+2*a*(b*x+a)*(1+(b*x+a)^2)^(1/2 
)/b^3/arcsinh(b*x+a)-(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b^3/arcsinh(b*x+a)-2*a* 
Chi(2*arcsinh(b*x+a))/b^3-1/4*Shi(arcsinh(b*x+a))/b^3+a^2*Shi(arcsinh(b*x+ 
a))/b^3+3/4*Shi(3*arcsinh(b*x+a))/b^3
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\frac {-\frac {4 b^2 x^2 \sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)}-8 a \text {Chi}(2 \text {arcsinh}(a+b x))+\left (-1+4 a^2\right ) \text {Shi}(\text {arcsinh}(a+b x))+3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \] Input:

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

Output:

((-4*b^2*x^2*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/ArcSinh[a + b*x] - 8*a*Cos 
hIntegral[2*ArcSinh[a + b*x]] + (-1 + 4*a^2)*SinhIntegral[ArcSinh[a + b*x] 
] + 3*SinhIntegral[3*ArcSinh[a + b*x]])/(4*b^3)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6274, 27, 6244, 2009}

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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6244

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))-\frac {a^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}-2 a \text {Chi}(2 \text {arcsinh}(a+b x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a+b x))+\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a+b x))+\frac {2 a (a+b x) \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}}{b^3}\)

Input:

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

Output:

(-((a^2*Sqrt[1 + (a + b*x)^2])/ArcSinh[a + b*x]) + (2*a*(a + b*x)*Sqrt[1 + 
 (a + b*x)^2])/ArcSinh[a + b*x] - ((a + b*x)^2*Sqrt[1 + (a + b*x)^2])/ArcS 
inh[a + b*x] - 2*a*CoshIntegral[2*ArcSinh[a + b*x]] - SinhIntegral[ArcSinh 
[a + b*x]]/4 + a^2*SinhIntegral[ArcSinh[a + b*x]] + (3*SinhIntegral[3*ArcS 
inh[a + b*x]])/4)/b^3
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6244

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

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 = 0.54 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {a \left (2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{\operatorname {arcsinh}\left (b x +a \right )}+\frac {a^{2} \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{3}}\)	\(146\)
default	\(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {a \left (2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{\operatorname {arcsinh}\left (b x +a \right )}+\frac {a^{2} \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{3}}\)	\(146\)

Input:

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

Output:

1/b^3*(1/4/arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)-1/4*Shi(arcsinh(b*x+a))-1/4/ 
arcsinh(b*x+a)*cosh(3*arcsinh(b*x+a))+3/4*Shi(3*arcsinh(b*x+a))-a*(2*Chi(2 
*arcsinh(b*x+a))*arcsinh(b*x+a)-sinh(2*arcsinh(b*x+a)))/arcsinh(b*x+a)+a^2 
*(Shi(arcsinh(b*x+a))*arcsinh(b*x+a)-(1+(b*x+a)^2)^(1/2))/arcsinh(b*x+a))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(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))/((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)*lo 
g(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))) + integrate((3*b^5*x^6 + 1 
4*a*b^4*x^5 + 2*(13*a^2*b^3 + 3*b^3)*x^4 + 8*(3*a^3*b^2 + 2*a*b^2)*x^3 + ( 
11*a^4*b + 14*a^2*b + 3*b)*x^2 + (3*b^3*x^4 + 8*a*b^2*x^3 + (7*a^2*b + b)* 
x^2 + 2*(a^3 + a)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*(a^5 + 2*a^3 + a)*x 
 + (6*b^4*x^5 + 22*a*b^3*x^4 + (30*a^2*b^2 + 7*b^2)*x^3 + (18*a^3*b + 13*a 
*b)*x^2 + 2*(2*a^4 + 3*a^2 + 1)*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((b^ 
5*x^4 + 4*a*b^4*x^3 + a^4*b + 2*a^2*b + 2*(3*a^2*b^3 + b^3)*x^2 + (b^3*x^2 
 + 2*a*b^2*x + a^2*b)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 4*(a^3*b^2 + a*b^2)* 
x + 2*(b^4*x^3 + 3*a*b^3*x^2 + a^3*b + a*b + (3*a^2*b^2 + b^2)*x)*sqrt(b^2 
*x^2 + 2*a*b*x + a^2 + 1) + b)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 
+ 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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