Integrand size = 10, antiderivative size = 155 \[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=-\frac {x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^3}-\frac {2 x^3}{3 a^2 \text {arcsinh}(a x)^2}-\frac {5 x^5}{6 \text {arcsinh}(a x)^2}-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a^3 \text {arcsinh}(a x)}-\frac {25 x^4 \sqrt {1+a^2 x^2}}{6 a \text {arcsinh}(a x)}+\frac {\text {Shi}(\text {arcsinh}(a x))}{48 a^5}-\frac {27 \text {Shi}(3 \text {arcsinh}(a x))}{32 a^5}+\frac {125 \text {Shi}(5 \text {arcsinh}(a x))}{96 a^5} \]
-2/3*x^3/a^2/arcsinh(a*x)^2-5/6*x^5/arcsinh(a*x)^2+1/48*Shi(arcsinh(a*x))/ a^5-27/32*Shi(3*arcsinh(a*x))/a^5+125/96*Shi(5*arcsinh(a*x))/a^5-1/3*x^4*( a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^3-2*x^2*(a^2*x^2+1)^(1/2)/a^3/arcsinh(a*x) -25/6*x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)
Time = 0.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=-\frac {32 a^4 x^4 \sqrt {1+a^2 x^2}+64 a^3 x^3 \text {arcsinh}(a x)+80 a^5 x^5 \text {arcsinh}(a x)+192 a^2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2+400 a^4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2-2 \text {arcsinh}(a x)^3 \text {Shi}(\text {arcsinh}(a x))+81 \text {arcsinh}(a x)^3 \text {Shi}(3 \text {arcsinh}(a x))-125 \text {arcsinh}(a x)^3 \text {Shi}(5 \text {arcsinh}(a x))}{96 a^5 \text {arcsinh}(a x)^3} \]
-1/96*(32*a^4*x^4*Sqrt[1 + a^2*x^2] + 64*a^3*x^3*ArcSinh[a*x] + 80*a^5*x^5 *ArcSinh[a*x] + 192*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 + 400*a^4*x^4 *Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 - 2*ArcSinh[a*x]^3*SinhIntegral[ArcSinh[ a*x]] + 81*ArcSinh[a*x]^3*SinhIntegral[3*ArcSinh[a*x]] - 125*ArcSinh[a*x]^ 3*SinhIntegral[5*ArcSinh[a*x]])/(a^5*ArcSinh[a*x]^3)
Time = 0.67 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.34, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6194, 6233, 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.
| \(\displaystyle \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {5}{3} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx+\frac {4 \int \frac {x^3}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}dx}{3 a}-\frac {x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {5}{3} a \left (\frac {5 \int \frac {x^4}{\text {arcsinh}(a x)^2}dx}{2 a}-\frac {x^5}{2 a \text {arcsinh}(a x)^2}\right )+\frac {4 \left (\frac {3 \int \frac {x^2}{\text {arcsinh}(a x)^2}dx}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {5}{3} a \left (\frac {5 \left (\frac {\int \left (\frac {a x}{8 \text {arcsinh}(a x)}-\frac {9 \sinh (3 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}+\frac {5 \sinh (5 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^5}{2 a \text {arcsinh}(a x)^2}\right )+\frac {4 \left (\frac {3 \left (\frac {\int \left (\frac {3 \sinh (3 \text {arcsinh}(a x))}{4 \text {arcsinh}(a x)}-\frac {a x}{4 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^3}+\frac {5}{3} a \left (\frac {5 \left (\frac {\frac {1}{8} \text {Shi}(\text {arcsinh}(a x))-\frac {9}{16} \text {Shi}(3 \text {arcsinh}(a x))+\frac {5}{16} \text {Shi}(5 \text {arcsinh}(a x))}{a^5}-\frac {x^4 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^5}{2 a \text {arcsinh}(a x)^2}\right )+\frac {4 \left (\frac {3 \left (\frac {\frac {3}{4} \text {Shi}(3 \text {arcsinh}(a x))-\frac {1}{4} \text {Shi}(\text {arcsinh}(a x))}{a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arcsinh}(a x)^2}\right )}{3 a}\) |
-1/3*(x^4*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x]^3) + (4*(-1/2*x^3/(a*ArcSinh[ a*x]^2) + (3*(-((x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + (-1/4*SinhInte gral[ArcSinh[a*x]] + (3*SinhIntegral[3*ArcSinh[a*x]])/4)/a^3))/(2*a)))/(3* a) + (5*a*(-1/2*x^5/(a*ArcSinh[a*x]^2) + (5*(-((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))/(2*a)))/3
3.1.67.3.1 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]
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]
Time = 0.06 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{24 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {a x}{48 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{48 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{48}+\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \operatorname {arcsinh}\left (a x \right )^{3}}+\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {9 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}-\frac {27 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{48 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {5 \sinh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{96 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {25 \cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{96 \,\operatorname {arcsinh}\left (a x \right )}+\frac {125 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{96}}{a^{5}}\) | \(169\) |
| default | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{24 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {a x}{48 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{48 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{48}+\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{16 \operatorname {arcsinh}\left (a x \right )^{3}}+\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {9 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}-\frac {27 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{48 \operatorname {arcsinh}\left (a x \right )^{3}}-\frac {5 \sinh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{96 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {25 \cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{96 \,\operatorname {arcsinh}\left (a x \right )}+\frac {125 \,\operatorname {Shi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{96}}{a^{5}}\) | \(169\) |
1/a^5*(-1/24/arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)-1/48*a*x/arcsinh(a*x)^2-1/48 /arcsinh(a*x)*(a^2*x^2+1)^(1/2)+1/48*Shi(arcsinh(a*x))+1/16/arcsinh(a*x)^3 *cosh(3*arcsinh(a*x))+3/32/arcsinh(a*x)^2*sinh(3*arcsinh(a*x))+9/32/arcsin h(a*x)*cosh(3*arcsinh(a*x))-27/32*Shi(3*arcsinh(a*x))-1/48/arcsinh(a*x)^3* cosh(5*arcsinh(a*x))-5/96/arcsinh(a*x)^2*sinh(5*arcsinh(a*x))-25/96/arcsin h(a*x)*cosh(5*arcsinh(a*x))+125/96*Shi(5*arcsinh(a*x)))
\[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
-1/6*(2*a^13*x^15 + 10*a^11*x^13 + 20*a^9*x^11 + 20*a^7*x^9 + 10*a^5*x^7 + 2*a^3*x^5 + 2*(a^8*x^10 + a^6*x^8)*(a^2*x^2 + 1)^(5/2) + 2*(5*a^9*x^11 + 9*a^7*x^9 + 4*a^5*x^7)*(a^2*x^2 + 1)^2 + (25*a^13*x^15 + 125*a^11*x^13 + 2 50*a^9*x^11 + 250*a^7*x^9 + 125*a^5*x^7 + 25*a^3*x^5 + (25*a^8*x^10 + 49*a ^6*x^8 + 27*a^4*x^6 + 3*a^2*x^4)*(a^2*x^2 + 1)^(5/2) + (125*a^9*x^11 + 321 *a^7*x^9 + 286*a^5*x^7 + 102*a^3*x^5 + 12*a*x^3)*(a^2*x^2 + 1)^2 + (250*a^ 10*x^12 + 794*a^8*x^10 + 946*a^6*x^8 + 519*a^4*x^6 + 129*a^2*x^4 + 12*x^2) *(a^2*x^2 + 1)^(3/2) + 2*(125*a^11*x^13 + 473*a^9*x^11 + 696*a^7*x^9 + 497 *a^5*x^7 + 173*a^3*x^5 + 24*a*x^3)*(a^2*x^2 + 1) + (125*a^12*x^14 + 549*a^ 10*x^12 + 955*a^8*x^10 + 824*a^6*x^8 + 354*a^4*x^6 + 61*a^2*x^4)*sqrt(a^2* x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))^2 + 4*(5*a^10*x^12 + 13*a^8*x^10 + 11*a^6*x^8 + 3*a^4*x^6)*(a^2*x^2 + 1)^(3/2) + 4*(5*a^11*x^13 + 17*a^9*x^11 + 21*a^7*x^9 + 11*a^5*x^7 + 2*a^3*x^5)*(a^2*x^2 + 1) + (5*a^13*x^15 + 25* a^11*x^13 + 50*a^9*x^11 + 50*a^7*x^9 + 25*a^5*x^7 + 5*a^3*x^5 + (5*a^8*x^1 0 + 8*a^6*x^8 + 3*a^4*x^6)*(a^2*x^2 + 1)^(5/2) + (25*a^9*x^11 + 57*a^7*x^9 + 42*a^5*x^7 + 10*a^3*x^5)*(a^2*x^2 + 1)^2 + (50*a^10*x^12 + 148*a^8*x^10 + 158*a^6*x^8 + 71*a^4*x^6 + 11*a^2*x^4)*(a^2*x^2 + 1)^(3/2) + 2*(25*a^11 *x^13 + 91*a^9*x^11 + 126*a^7*x^9 + 81*a^5*x^7 + 23*a^3*x^5 + 2*a*x^3)*(a^ 2*x^2 + 1) + (25*a^12*x^14 + 108*a^10*x^12 + 183*a^8*x^10 + 151*a^6*x^8 + 60*a^4*x^6 + 9*a^2*x^4)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))...
\[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^4} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^4} \,d x \]