Integrand size = 10, antiderivative size = 173 \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\frac {5^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-5 \text {arcsinh}(a x))}{32 a^5}-\frac {3^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{32 a^5}+\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{16 a^5}-\frac {\Gamma (1+n,\text {arcsinh}(a x))}{16 a^5}+\frac {3^{-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{32 a^5}-\frac {5^{-1-n} \Gamma (1+n,5 \text {arcsinh}(a x))}{32 a^5} \]
1/32*5^(-1-n)*arcsinh(a*x)^n*GAMMA(1+n,-5*arcsinh(a*x))/a^5/((-arcsinh(a*x ))^n)-1/32*arcsinh(a*x)^n*GAMMA(1+n,-3*arcsinh(a*x))/(3^n)/a^5/((-arcsinh( a*x))^n)+1/16*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/a^5/((-arcsinh(a*x)) ^n)-1/16*GAMMA(1+n,arcsinh(a*x))/a^5+1/32*GAMMA(1+n,3*arcsinh(a*x))/(3^n)/ a^5-1/32*5^(-1-n)*GAMMA(1+n,5*arcsinh(a*x))/a^5
Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.84 \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\frac {5^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-5 \text {arcsinh}(a x))-5\ 3^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))+10 (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))-10 \Gamma (1+n,\text {arcsinh}(a x))+5\ 3^{-n} \Gamma (1+n,3 \text {arcsinh}(a x))-5^{-n} \Gamma (1+n,5 \text {arcsinh}(a x))}{160 a^5} \]
((ArcSinh[a*x]^n*Gamma[1 + n, -5*ArcSinh[a*x]])/(5^n*(-ArcSinh[a*x])^n) - (5*ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(3^n*(-ArcSinh[a*x])^n) + (10*ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(-ArcSinh[a*x])^n - 10*Ga mma[1 + n, ArcSinh[a*x]] + (5*Gamma[1 + n, 3*ArcSinh[a*x]])/3^n - Gamma[1 + n, 5*ArcSinh[a*x]]/5^n)/(160*a^5)
Time = 0.43 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {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.
\(\displaystyle \int x^4 \text {arcsinh}(a x)^n \, dx\) |
\(\Big \downarrow \) 6195 |
\(\displaystyle \frac {\int a^4 x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^nd\text {arcsinh}(a x)}{a^5}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {\int \left (-\frac {3}{16} \cosh (3 \text {arcsinh}(a x)) \text {arcsinh}(a x)^n+\frac {1}{16} \cosh (5 \text {arcsinh}(a x)) \text {arcsinh}(a x)^n+\frac {1}{8} \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^n\right )d\text {arcsinh}(a x)}{a^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{32} 5^{-n-1} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-5 \text {arcsinh}(a x))-\frac {1}{32} 3^{-n} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-3 \text {arcsinh}(a x))+\frac {1}{16} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-\text {arcsinh}(a x))-\frac {1}{16} \Gamma (n+1,\text {arcsinh}(a x))+\frac {1}{32} 3^{-n} \Gamma (n+1,3 \text {arcsinh}(a x))-\frac {1}{32} 5^{-n-1} \Gamma (n+1,5 \text {arcsinh}(a x))}{a^5}\) |
((5^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -5*ArcSinh[a*x]])/(32*(-ArcSinh[a *x])^n) - (ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(32*3^n*(-ArcSinh [a*x])^n) + (ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(16*(-ArcSinh[a*x ])^n) - Gamma[1 + n, ArcSinh[a*x]]/16 + Gamma[1 + n, 3*ArcSinh[a*x]]/(32*3 ^n) - (5^(-1 - n)*Gamma[1 + n, 5*ArcSinh[a*x]])/32)/a^5
3.2.29.3.1 Defintions of rubi rules used
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_)^(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]
\[\int x^{4} \operatorname {arcsinh}\left (a x \right )^{n}d x\]
\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int x^{4} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]
\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
Timed out. \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]