Integrand size = 10, antiderivative size = 81 \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \]
-x/a^2/arcsinh(a*x)-3/2*x^3/arcsinh(a*x)-1/8*Chi(arcsinh(a*x))/a^3+9/8*Chi (3*arcsinh(a*x))/a^3-1/2*x^2*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^2
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {\frac {4 a x \left (a x \sqrt {1+a^2 x^2}+\left (2+3 a^2 x^2\right ) \text {arcsinh}(a x)\right )}{\text {arcsinh}(a x)^2}+\text {Chi}(\text {arcsinh}(a x))-9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \]
-1/8*((4*a*x*(a*x*Sqrt[1 + a^2*x^2] + (2 + 3*a^2*x^2)*ArcSinh[a*x]))/ArcSi nh[a*x]^2 + CoshIntegral[ArcSinh[a*x]] - 9*CoshIntegral[3*ArcSinh[a*x]])/a ^3
Time = 0.83 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6194, 6233, 6189, 3042, 3782, 6195, 5971, 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.
| \(\displaystyle \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {\int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}dx}{a}+\frac {3}{2} a \int \frac {x^3}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}dx-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {3}{2} a \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)}dx}{a}-\frac {x^3}{a \text {arcsinh}(a x)}\right )+\frac {\frac {\int \frac {1}{\text {arcsinh}(a x)}dx}{a}-\frac {x}{a \text {arcsinh}(a x)}}{a}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 6189 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a^2 x^2+1}}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^2}-\frac {x}{a \text {arcsinh}(a x)}}{a}+\frac {3}{2} a \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)}dx}{a}-\frac {x^3}{a \text {arcsinh}(a x)}\right )-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {x}{a \text {arcsinh}(a x)}+\frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^2}}{a}+\frac {3}{2} a \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)}dx}{a}-\frac {x^3}{a \text {arcsinh}(a x)}\right )-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {3}{2} a \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)}dx}{a}-\frac {x^3}{a \text {arcsinh}(a x)}\right )+\frac {\frac {\text {Chi}(\text {arcsinh}(a x))}{a^2}-\frac {x}{a \text {arcsinh}(a x)}}{a}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {3}{2} a \left (\frac {3 \int \frac {a^2 x^2 \sqrt {a^2 x^2+1}}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a^4}-\frac {x^3}{a \text {arcsinh}(a x)}\right )+\frac {\frac {\text {Chi}(\text {arcsinh}(a x))}{a^2}-\frac {x}{a \text {arcsinh}(a x)}}{a}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {3}{2} a \left (\frac {3 \int \left (\frac {\cosh (3 \text {arcsinh}(a x))}{4 \text {arcsinh}(a x)}-\frac {\sqrt {a^2 x^2+1}}{4 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^4}-\frac {x^3}{a \text {arcsinh}(a x)}\right )+\frac {\frac {\text {Chi}(\text {arcsinh}(a x))}{a^2}-\frac {x}{a \text {arcsinh}(a x)}}{a}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} a \left (\frac {3 \left (\frac {1}{4} \text {Chi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Chi}(\text {arcsinh}(a x))\right )}{a^4}-\frac {x^3}{a \text {arcsinh}(a x)}\right )+\frac {\frac {\text {Chi}(\text {arcsinh}(a x))}{a^2}-\frac {x}{a \text {arcsinh}(a x)}}{a}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}\) |
-1/2*(x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^2) + (-(x/(a*ArcSinh[a*x])) + CoshIntegral[ArcSinh[a*x]]/a^2)/a + (3*a*(-(x^3/(a*ArcSinh[a*x])) + (3*(- 1/4*CoshIntegral[ArcSinh[a*x]] + CoshIntegral[3*ArcSinh[a*x]]/4))/a^4))/2
3.1.62.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {a x}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(81\) |
| default | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {a x}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(81\) |
1/a^3*(1/8/arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)+1/8*a*x/arcsinh(a*x)-1/8*Chi(a rcsinh(a*x))-1/8/arcsinh(a*x)^2*cosh(3*arcsinh(a*x))-3/8/arcsinh(a*x)*sinh (3*arcsinh(a*x))+9/8*Chi(3*arcsinh(a*x)))
\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
-1/2*(a^8*x^9 + 3*a^6*x^7 + 3*a^4*x^5 + a^2*x^3 + (a^5*x^6 + a^3*x^4)*(a^2 *x^2 + 1)^(3/2) + (3*a^6*x^7 + 5*a^4*x^5 + 2*a^2*x^3)*(a^2*x^2 + 1) + (3*a ^8*x^9 + 9*a^6*x^7 + 9*a^4*x^5 + 3*a^2*x^3 + (3*a^5*x^6 + 4*a^3*x^4 + a*x^ 2)*(a^2*x^2 + 1)^(3/2) + (9*a^6*x^7 + 17*a^4*x^5 + 10*a^2*x^3 + 2*x)*(a^2* x^2 + 1) + (9*a^7*x^8 + 22*a^5*x^6 + 18*a^3*x^4 + 5*a*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + (3*a^7*x^8 + 7*a^5*x^6 + 5*a^3*x^4 + a* x^2)*sqrt(a^2*x^2 + 1))/((a^8*x^6 + 3*a^6*x^4 + (a^2*x^2 + 1)^(3/2)*a^5*x^ 3 + 3*a^4*x^2 + 3*(a^6*x^4 + a^4*x^2)*(a^2*x^2 + 1) + a^2 + 3*(a^7*x^5 + 2 *a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2) + int egrate(1/2*(9*a^10*x^10 + 36*a^8*x^8 + 54*a^6*x^6 + 36*a^4*x^4 + 9*a^2*x^2 + (9*a^6*x^6 + 4*a^4*x^4 - a^2*x^2)*(a^2*x^2 + 1)^2 + (36*a^7*x^7 + 48*a^ 5*x^5 + 13*a^3*x^3 - 2*a*x)*(a^2*x^2 + 1)^(3/2) + (54*a^8*x^8 + 120*a^6*x^ 6 + 83*a^4*x^4 + 19*a^2*x^2 + 2)*(a^2*x^2 + 1) + (36*a^9*x^9 + 112*a^7*x^7 + 123*a^5*x^5 + 57*a^3*x^3 + 10*a*x)*sqrt(a^2*x^2 + 1))/((a^10*x^8 + 4*a^ 8*x^6 + (a^2*x^2 + 1)^2*a^6*x^4 + 6*a^6*x^4 + 4*a^4*x^2 + 4*(a^7*x^5 + a^5 *x^3)*(a^2*x^2 + 1)^(3/2) + 6*(a^8*x^6 + 2*a^6*x^4 + a^4*x^2)*(a^2*x^2 + 1 ) + a^2 + 4*(a^9*x^7 + 3*a^7*x^5 + 3*a^5*x^3 + a^3*x)*sqrt(a^2*x^2 + 1))*l og(a*x + sqrt(a^2*x^2 + 1))), x)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]