Integrand size = 10, antiderivative size = 68 \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=-\frac {x^4 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}+\frac {\text {Shi}(\text {arcsinh}(a x))}{8 a^5}-\frac {9 \text {Shi}(3 \text {arcsinh}(a x))}{16 a^5}+\frac {5 \text {Shi}(5 \text {arcsinh}(a x))}{16 a^5} \]
1/8*Shi(arcsinh(a*x))/a^5-9/16*Shi(3*arcsinh(a*x))/a^5+5/16*Shi(5*arcsinh( a*x))/a^5-x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)
Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\frac {-\frac {16 a^4 x^4 \sqrt {1+a^2 x^2}}{\text {arcsinh}(a x)}+2 \text {Shi}(\text {arcsinh}(a x))-9 \text {Shi}(3 \text {arcsinh}(a x))+5 \text {Shi}(5 \text {arcsinh}(a x))}{16 a^5} \]
((-16*a^4*x^4*Sqrt[1 + a^2*x^2])/ArcSinh[a*x] + 2*SinhIntegral[ArcSinh[a*x ]] - 9*SinhIntegral[3*ArcSinh[a*x]] + 5*SinhIntegral[5*ArcSinh[a*x]])/(16* a^5)
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6193, 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.
\(\displaystyle \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx\) |
\(\Big \downarrow \) 6193 |
\(\displaystyle \frac {\int \left (\frac {a x}{8 \text {arcsinh}(a x)}-\frac {9 \sinh (3 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}+\frac {5 \sinh (5 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{8} \text {Shi}(\text {arcsinh}(a x))-\frac {9}{16} \text {Shi}(3 \text {arcsinh}(a x))+\frac {5}{16} \text {Shi}(5 \text {arcsinh}(a x))}{a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\) |
-((x^4*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + (SinhIntegral[ArcSinh[a*x]]/ 8 - (9*SinhIntegral[3*ArcSinh[a*x]])/16 + (5*SinhIntegral[5*ArcSinh[a*x]]) /16)/a^5
3.1.53.3.1 Defintions of rubi rules used
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]
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}+\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}-\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}+\frac {5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{5}}\) | \(80\) |
default | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}+\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}-\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \,\operatorname {arcsinh}\left (a x \right )}+\frac {5 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{16}}{a^{5}}\) | \(80\) |
1/a^5*(-1/8/arcsinh(a*x)*(a^2*x^2+1)^(1/2)+1/8*Shi(arcsinh(a*x))+3/16/arcs inh(a*x)*cosh(3*arcsinh(a*x))-9/16*Shi(3*arcsinh(a*x))-1/16/arcsinh(a*x)*c osh(5*arcsinh(a*x))+5/16*Shi(5*arcsinh(a*x)))
\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
-(a^3*x^7 + a*x^5 + (a^2*x^6 + x^4)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^ 2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a^2*x^2 + 1))) + integrate((5*a^5*x^8 + 10*a^3*x^6 + 5*a*x^4 + (5*a^3*x^6 + 3*a*x^4)*(a^2*x^2 + 1) + (10*a^4*x^ 7 + 13*a^2*x^5 + 4*x^3)*sqrt(a^2*x^2 + 1))/((a^5*x^4 + (a^2*x^2 + 1)*a^3*x ^2 + 2*a^3*x^2 + 2*(a^4*x^3 + a^2*x)*sqrt(a^2*x^2 + 1) + a)*log(a*x + sqrt (a^2*x^2 + 1))), x)
\[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]