Integrand size = 10, antiderivative size = 84 \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\frac {a \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \text {arcsinh}(a+b x)}+\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b^2}-\frac {a \text {Shi}(\text {arcsinh}(a+b x))}{b^2} \]
Chi(2*arcsinh(b*x+a))/b^2-a*Shi(arcsinh(b*x+a))/b^2+a*(1+(b*x+a)^2)^(1/2)/ b^2/arcsinh(b*x+a)-(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^2/arcsinh(b*x+a)
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {b x \sqrt {1+(a+b x)^2}-\text {arcsinh}(a+b x) \text {Chi}(2 \text {arcsinh}(a+b x))+a \text {arcsinh}(a+b x) \text {Shi}(\text {arcsinh}(a+b x))}{b^2 \text {arcsinh}(a+b x)} \]
-((b*x*Sqrt[1 + (a + b*x)^2] - ArcSinh[a + b*x]*CoshIntegral[2*ArcSinh[a + b*x]] + a*ArcSinh[a + b*x]*SinhIntegral[ArcSinh[a + b*x]])/(b^2*ArcSinh[a + b*x]))
Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6274, 25, 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.
\(\displaystyle \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int \frac {x}{\text {arcsinh}(a+b x)^2}d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -\frac {x}{\text {arcsinh}(a+b x)^2}d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -\frac {b x}{\text {arcsinh}(a+b x)^2}d(a+b x)}{b^2}\) |
\(\Big \downarrow \) 6244 |
\(\displaystyle -\frac {\int \left (\frac {a}{\text {arcsinh}(a+b x)^2}-\frac {a+b x}{\text {arcsinh}(a+b x)^2}\right )d(a+b x)}{b^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\text {Chi}(2 \text {arcsinh}(a+b x))+a \text {Shi}(\text {arcsinh}(a+b x))-\frac {a \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}+\frac {(a+b x) \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}}{b^2}\) |
-((-((a*Sqrt[1 + (a + b*x)^2])/ArcSinh[a + b*x]) + ((a + b*x)*Sqrt[1 + (a + b*x)^2])/ArcSinh[a + b*x] - CoshIntegral[2*ArcSinh[a + b*x]] + a*SinhInt egral[ArcSinh[a + b*x]])/b^2)
3.1.86.3.1 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 = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 \,\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\frac {a \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{2}}\) | \(73\) |
default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 \,\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\frac {a \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{2}}\) | \(73\) |
1/b^2*(-1/2/arcsinh(b*x+a)*sinh(2*arcsinh(b*x+a))+Chi(2*arcsinh(b*x+a))-a* (Shi(arcsinh(b*x+a))*arcsinh(b*x+a)-(1+(b*x+a)^2)^(1/2))/arcsinh(b*x+a))
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
-(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))/((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))) + integrate((2*b^5*x^5 + 9*a*b ^4*x^4 + a^5 + 4*(4*a^2*b^3 + b^3)*x^3 + 2*a^3 + 2*(7*a^3*b^2 + 5*a*b^2)*x ^2 + (2*b^3*x^3 + 5*a*b^2*x^2 + 4*a^2*b*x + a^3 + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*(3*a^4*b + 4*a^2*b + b)*x + (4*b^4*x^4 + 14*a*b^3*x^3 + 2*a^4 + 2*(9*a^2*b^2 + 2*b^2)*x^2 + 3*a^2 + (10*a^3*b + 7*a*b)*x + 1)*sqrt(b^2* x^2 + 2*a*b*x + a^2 + 1) + a)/((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)
\[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]