Integrand size = 10, antiderivative size = 138 \[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=-\frac {x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {x}{3 a^2 \text {arcsinh}(a x)^2}-\frac {x^3}{2 \text {arcsinh}(a x)^2}-\frac {\sqrt {1+a^2 x^2}}{3 a^3 \text {arcsinh}(a x)}-\frac {3 x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)}-\frac {\text {Shi}(\text {arcsinh}(a x))}{24 a^3}+\frac {9 \text {Shi}(3 \text {arcsinh}(a x))}{8 a^3} \]
-1/3*x/a^2/arcsinh(a*x)^2-1/2*x^3/arcsinh(a*x)^2-1/24*Shi(arcsinh(a*x))/a^ 3+9/8*Shi(3*arcsinh(a*x))/a^3-1/3*x^2*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^3-1 /3*(a^2*x^2+1)^(1/2)/a^3/arcsinh(a*x)-3/2*x^2*(a^2*x^2+1)^(1/2)/a/arcsinh( a*x)
Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.72 \[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=-\frac {\frac {4 \left (2 a^2 x^2 \sqrt {1+a^2 x^2}+a x \left (2+3 a^2 x^2\right ) \text {arcsinh}(a x)+\sqrt {1+a^2 x^2} \left (2+9 a^2 x^2\right ) \text {arcsinh}(a x)^2\right )}{\text {arcsinh}(a x)^3}+\text {Shi}(\text {arcsinh}(a x))-27 \text {Shi}(3 \text {arcsinh}(a x))}{24 a^3} \]
-1/24*((4*(2*a^2*x^2*Sqrt[1 + a^2*x^2] + a*x*(2 + 3*a^2*x^2)*ArcSinh[a*x] + Sqrt[1 + a^2*x^2]*(2 + 9*a^2*x^2)*ArcSinh[a*x]^2))/ArcSinh[a*x]^3 + Sinh Integral[ArcSinh[a*x]] - 27*SinhIntegral[3*ArcSinh[a*x]])/a^3
Time = 1.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6194, 6233, 6188, 6193, 2009, 6234, 3042, 26, 3779}
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.
| \(\displaystyle \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {2 \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx}{3 a}+a \int \frac {x^3}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle a \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)^2}dx}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )+\frac {2 \left (\frac {\int \frac {1}{\text {arcsinh}(a x)^2}dx}{2 a}-\frac {x}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {2 \left (\frac {a \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}dx-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{2 a}-\frac {x}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}+a \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)^2}dx}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {2 \left (\frac {a \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}dx-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{2 a}-\frac {x}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}+a \left (\frac {3 \left (\frac {\int \left (\frac {3 \sinh (3 \text {arcsinh}(a x))}{4 \text {arcsinh}(a x)}-\frac {a x}{4 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {a \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}dx-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{2 a}-\frac {x}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+a \left (\frac {3 \left (\frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {a x}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{2 a}-\frac {x}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+a \left (\frac {3 \left (\frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (-\frac {x}{2 a \text {arcsinh}(a x)^2}+\frac {-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}+\frac {\int -\frac {i \sin (i \text {arcsinh}(a x))}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}}{2 a}\right )}{3 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+a \left (\frac {3 \left (\frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 \left (-\frac {x}{2 a \text {arcsinh}(a x)^2}+\frac {-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}-\frac {i \int \frac {\sin (i \text {arcsinh}(a x))}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}}{2 a}\right )}{3 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+a \left (\frac {3 \left (\frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\text {Shi}(\text {arcsinh}(a x))}{a}-\frac {\sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}}{2 a}-\frac {x}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+a \left (\frac {3 \left (\frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )\) |
-1/3*(x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^3) + (2*(-1/2*x/(a*ArcSinh[a* x]^2) + (-(Sqrt[1 + a^2*x^2]/(a*ArcSinh[a*x])) + SinhIntegral[ArcSinh[a*x] ]/a)/(2*a)))/(3*a) + a*(-1/2*x^3/(a*ArcSinh[a*x]^2) + (3*(-((x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + (-1/4*SinhIntegral[ArcSinh[a*x]] + (3*SinhIn tegral[3*ArcSinh[a*x]])/4)/a^3))/(2*a))
3.1.69.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
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] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
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_)*((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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{12 \operatorname {arcsinh}\left (a x \right )^{3}}+\frac {a x}{24 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {\sqrt {a^{2} x^{2}+1}}{24 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{24}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{12 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {\sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(115\) |
| default | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{12 \operatorname {arcsinh}\left (a x \right )^{3}}+\frac {a x}{24 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {\sqrt {a^{2} x^{2}+1}}{24 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{24}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{12 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {\sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(115\) |
1/a^3*(1/12/arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)+1/24*a*x/arcsinh(a*x)^2+1/24/ arcsinh(a*x)*(a^2*x^2+1)^(1/2)-1/24*Shi(arcsinh(a*x))-1/12/arcsinh(a*x)^3* cosh(3*arcsinh(a*x))-1/8/arcsinh(a*x)^2*sinh(3*arcsinh(a*x))-3/8/arcsinh(a *x)*cosh(3*arcsinh(a*x))+9/8*Shi(3*arcsinh(a*x)))
\[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
-1/6*(2*a^13*x^13 + 10*a^11*x^11 + 20*a^9*x^9 + 20*a^7*x^7 + 10*a^5*x^5 + 2*a^3*x^3 + 2*(a^8*x^8 + a^6*x^6)*(a^2*x^2 + 1)^(5/2) + 2*(5*a^9*x^9 + 9*a ^7*x^7 + 4*a^5*x^5)*(a^2*x^2 + 1)^2 + (9*a^13*x^13 + 45*a^11*x^11 + 90*a^9 *x^9 + 90*a^7*x^7 + 45*a^5*x^5 + 9*a^3*x^3 + (9*a^8*x^8 + 13*a^6*x^6 + 3*a ^4*x^4 - a^2*x^2)*(a^2*x^2 + 1)^(5/2) + (45*a^9*x^9 + 97*a^7*x^7 + 64*a^5* x^5 + 10*a^3*x^3 - 2*a*x)*(a^2*x^2 + 1)^2 + (90*a^10*x^10 + 258*a^8*x^8 + 264*a^6*x^6 + 113*a^4*x^4 + 19*a^2*x^2 + 2)*(a^2*x^2 + 1)^(3/2) + 2*(45*a^ 11*x^11 + 161*a^9*x^9 + 219*a^7*x^7 + 141*a^5*x^5 + 44*a^3*x^3 + 6*a*x)*(a ^2*x^2 + 1) + (45*a^12*x^12 + 193*a^10*x^10 + 325*a^8*x^8 + 270*a^6*x^6 + 112*a^4*x^4 + 19*a^2*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^ 2 + 4*(5*a^10*x^10 + 13*a^8*x^8 + 11*a^6*x^6 + 3*a^4*x^4)*(a^2*x^2 + 1)^(3 /2) + 4*(5*a^11*x^11 + 17*a^9*x^9 + 21*a^7*x^7 + 11*a^5*x^5 + 2*a^3*x^3)*( a^2*x^2 + 1) + (3*a^13*x^13 + 15*a^11*x^11 + 30*a^9*x^9 + 30*a^7*x^7 + 15* a^5*x^5 + 3*a^3*x^3 + (3*a^8*x^8 + 4*a^6*x^6 + a^4*x^4)*(a^2*x^2 + 1)^(5/2 ) + (15*a^9*x^9 + 31*a^7*x^7 + 20*a^5*x^5 + 4*a^3*x^3)*(a^2*x^2 + 1)^2 + ( 30*a^10*x^10 + 84*a^8*x^8 + 84*a^6*x^6 + 35*a^4*x^4 + 5*a^2*x^2)*(a^2*x^2 + 1)^(3/2) + 2*(15*a^11*x^11 + 53*a^9*x^9 + 71*a^7*x^7 + 44*a^5*x^5 + 12*a ^3*x^3 + a*x)*(a^2*x^2 + 1) + (15*a^12*x^12 + 64*a^10*x^10 + 107*a^8*x^8 + 87*a^6*x^6 + 34*a^4*x^4 + 5*a^2*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^ 2*x^2 + 1)) + 2*(5*a^12*x^12 + 21*a^10*x^10 + 34*a^8*x^8 + 26*a^6*x^6 +...
\[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^4} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^4} \,d x \]