Integrand size = 10, antiderivative size = 87 \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}+\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^4}+\frac {\text {Shi}(4 \text {arccosh}(a x))}{a^4} \]
3/2*x^2/a^2/arccosh(a*x)-2*x^4/arccosh(a*x)+1/2*Shi(2*arccosh(a*x))/a^4+Sh i(4*arccosh(a*x))/a^4-1/2*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^2
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\frac {-\frac {a^2 x^2 \left (a x \sqrt {-1+a x} \sqrt {1+a x}+\left (-3+4 a^2 x^2\right ) \text {arccosh}(a x)\right )}{\text {arccosh}(a x)^2}+\text {Shi}(2 \text {arccosh}(a x))+2 \text {Shi}(4 \text {arccosh}(a x))}{2 a^4} \]
(-((a^2*x^2*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + (-3 + 4*a^2*x^2)*ArcCosh[a *x]))/ArcCosh[a*x]^2) + SinhIntegral[2*ArcCosh[a*x]] + 2*SinhIntegral[4*Ar cCosh[a*x]])/(2*a^4)
Time = 1.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6301, 6366, 6302, 5971, 27, 2009, 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.
| \(\displaystyle \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle 2 a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}dx-\frac {3 \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}dx}{2 a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle -\frac {3 \left (\frac {2 \int \frac {x}{\text {arccosh}(a x)}dx}{a}-\frac {x^2}{a \text {arccosh}(a x)}\right )}{2 a}+2 a \left (\frac {4 \int \frac {x^3}{\text {arccosh}(a x)}dx}{a}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle -\frac {3 \left (\frac {2 \int \frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3}-\frac {x^2}{a \text {arccosh}(a x)}\right )}{2 a}+2 a \left (\frac {4 \int \frac {a^3 x^3 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle 2 a \left (\frac {4 \int \left (\frac {\sinh (2 \text {arccosh}(a x))}{4 \text {arccosh}(a x)}+\frac {\sinh (4 \text {arccosh}(a x))}{8 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {3 \left (\frac {2 \int \frac {\sinh (2 \text {arccosh}(a x))}{2 \text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3}-\frac {x^2}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a \left (\frac {4 \int \left (\frac {\sinh (2 \text {arccosh}(a x))}{4 \text {arccosh}(a x)}+\frac {\sinh (4 \text {arccosh}(a x))}{8 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {3 \left (\frac {\int \frac {\sinh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3}-\frac {x^2}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {\sinh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3}-\frac {x^2}{a \text {arccosh}(a x)}\right )}{2 a}+2 a \left (\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arccosh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arccosh}(a x))\right )}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {x^2}{a \text {arccosh}(a x)}+\frac {\int -\frac {i \sin (2 i \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3}\right )}{2 a}+2 a \left (\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arccosh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arccosh}(a x))\right )}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {3 \left (-\frac {x^2}{a \text {arccosh}(a x)}-\frac {i \int \frac {\sin (2 i \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^3}\right )}{2 a}+2 a \left (\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arccosh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arccosh}(a x))\right )}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle 2 a \left (\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arccosh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arccosh}(a x))\right )}{a^5}-\frac {x^4}{a \text {arccosh}(a x)}\right )-\frac {3 \left (\frac {\text {Shi}(2 \text {arccosh}(a x))}{a^3}-\frac {x^2}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
-1/2*(x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) - (3*(-(x^2/(a* ArcCosh[a*x])) + SinhIntegral[2*ArcCosh[a*x]]/a^3))/(2*a) + 2*a*(-(x^4/(a* ArcCosh[a*x])) + (4*(SinhIntegral[2*ArcCosh[a*x]]/4 + SinhIntegral[4*ArcCo sh[a*x]]/8))/a^5)
3.1.59.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{4}}\) | \(82\) |
| default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{4}}\) | \(82\) |
1/a^4*(-1/8/arccosh(a*x)^2*sinh(2*arccosh(a*x))-1/4/arccosh(a*x)*cosh(2*ar ccosh(a*x))+1/2*Shi(2*arccosh(a*x))-1/16/arccosh(a*x)^2*sinh(4*arccosh(a*x ))-1/4/arccosh(a*x)*cosh(4*arccosh(a*x))+Shi(4*arccosh(a*x)))
\[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
\[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int \frac {x^{3}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
-1/2*(a^8*x^10 - 3*a^6*x^8 + 3*a^4*x^6 - a^2*x^4 + (a^5*x^7 - a^3*x^5)*(a* x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^6*x^8 - 5*a^4*x^6 + 2*a^2*x^4)*(a*x + 1)*(a*x - 1) + (3*a^7*x^9 - 7*a^5*x^7 + 5*a^3*x^5 - a*x^3)*sqrt(a*x + 1)*s qrt(a*x - 1) + (4*a^8*x^10 - 12*a^6*x^8 + 12*a^4*x^6 - 4*a^2*x^4 + 2*(2*a^ 5*x^7 - 3*a^3*x^5 + a*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 3*(4*a^6*x^8 - 8*a^4*x^6 + 5*a^2*x^4 - x^2)*(a*x + 1)*(a*x - 1) + (12*a^7*x^9 - 30*a^5* x^7 + 25*a^3*x^5 - 7*a*x^3)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a* x + 1)*sqrt(a*x - 1)))/((a^8*x^6 + (a*x + 1)^(3/2)*(a*x - 1)^(3/2)*a^5*x^3 - 3*a^6*x^4 + 3*a^4*x^2 + 3*(a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x - 1) + 3*( a^7*x^5 - 2*a^5*x^3 + a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2) + integrate(1/2*(16*a^10*x^11 - 64*a^8*x^9 + 96*a^6*x^7 - 64*a^4*x^5 + 4*(4*a^6*x^7 - 3*a^4*x^5)*(a*x + 1)^2*(a*x - 1)^2 + 16*a^2*x^3 + (64*a^7*x^8 - 100*a^5*x^6 + 42*a^3*x^4 - 3*a*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(16*a^8*x^9 - 38*a^6*x^7 + 30*a^4*x^5 - 9* a^2*x^3 + x)*(a*x + 1)*(a*x - 1) + (64*a^9*x^10 - 204*a^7*x^8 + 234*a^5*x^ 6 - 115*a^3*x^4 + 21*a*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^10*x^8 + (a*x + 1)^2*(a*x - 1)^2*a^6*x^4 - 4*a^8*x^6 + 6*a^6*x^4 - 4*a^4*x^2 + 4*(a^7*x ^5 - a^5*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^8*x^6 - 2*a^6*x^4 + a ^4*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^9*x^7 - 3*a^7*x^5 + 3*a^5*x^3 - a^3*x)* sqrt(a*x + 1)*sqrt(a*x - 1) + a^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1...
Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]