Integrand size = 12, antiderivative size = 154 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {2 a \text {Chi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {\text {Shi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))}{b^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \]
-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-a^2*(1+(b*x+a)^2)^(1/2)/b^3/arc sinh(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)
Time = 0.77 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\frac {-\frac {4 b^2 x^2 \sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)}-8 a \text {Chi}(2 \text {arcsinh}(a+b x))+\left (-1+4 a^2\right ) \text {Shi}(\text {arcsinh}(a+b x))+3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \]
((-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)
Time = 0.42 (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 |
\(\displaystyle \frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))-\frac {a^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}-2 a \text {Chi}(2 \text {arcsinh}(a+b x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a+b x))+\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a+b x))+\frac {2 a (a+b x) \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}}{b^3}\) |
(-((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
3.1.85.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Time = 0.28 (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 | \(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {a \left (2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{\operatorname {arcsinh}\left (b x +a \right )}+\frac {a^{2} \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{3}}\) | \(146\) |
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))
\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
-(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)
\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]