Integrand size = 6, antiderivative size = 55 \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {x}{2 \text {arccosh}(a x)}+\frac {\text {Shi}(\text {arccosh}(a x))}{2 a} \]
-1/2*x/arccosh(a*x)+1/2*Shi(arccosh(a*x))/a-1/2*(a*x-1)^(1/2)*(a*x+1)^(1/2 )/a/arccosh(a*x)^2
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {x}{2 \text {arccosh}(a x)}+\frac {\text {Shi}(\text {arccosh}(a x))}{2 a} \]
-1/2*(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) - x/(2*ArcCosh[a*x] ) + SinhIntegral[ArcCosh[a*x]]/(2*a)
Time = 0.54 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6295, 6366, 6296, 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.
\(\displaystyle \int \frac {1}{\text {arccosh}(a x)^3} \, dx\) |
\(\Big \downarrow \) 6295 |
\(\displaystyle \frac {1}{2} a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
\(\Big \downarrow \) 6366 |
\(\displaystyle \frac {1}{2} a \left (\frac {\int \frac {1}{\text {arccosh}(a x)}dx}{a}-\frac {x}{a \text {arccosh}(a x)}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
\(\Big \downarrow \) 6296 |
\(\displaystyle \frac {1}{2} a \left (\frac {\int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x}{a \text {arccosh}(a x)}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2} a \left (-\frac {x}{a \text {arccosh}(a x)}+\frac {\int -\frac {i \sin (i \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2} a \left (-\frac {x}{a \text {arccosh}(a x)}-\frac {i \int \frac {\sin (i \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {1}{2} a \left (\frac {\text {Shi}(\text {arccosh}(a x))}{a^2}-\frac {x}{a \text {arccosh}(a x)}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2}\) |
-1/2*(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) + (a*(-(x/(a*ArcCos h[a*x])) + SinhIntegral[ArcCosh[a*x]]/a^2))/2
3.1.62.3.1 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_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c* x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c /(b*(n + 1)) Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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 = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {a x}{2 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{2}}{a}\) | \(45\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {a x}{2 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{2}}{a}\) | \(45\) |
1/a*(-1/2/arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/2*a*x/arccosh(a*x)+ 1/2*Shi(arccosh(a*x)))
\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int \frac {1}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
-1/2*(a^7*x^7 - 3*a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 - a^2*x^2)*(a*x + 1)^(3/2 )*(a*x - 1)^(3/2) + (3*a^5*x^5 - 5*a^3*x^3 + 2*a*x)*(a*x + 1)*(a*x - 1) + (3*a^6*x^6 - 7*a^4*x^4 + 5*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x + (a^7*x^7 - 3*a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 - 1)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 3*(a^5*x^5 - a^3*x^3)*(a*x + 1)*(a*x - 1) + (3*a^6*x^6 - 6*a^4* x^4 + 4*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^7*x^6 + (a*x + 1)^(3/2)*(a*x - 1)^(3/2)*a^4*x^3 - 3*a^5*x^4 + 3*a^3*x^2 + 3*(a^5*x^4 - a^3*x^2)*(a*x + 1)*(a*x - 1) + 3*(a ^6*x^5 - 2*a^4*x^3 + a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a)*log(a*x + sqr t(a*x + 1)*sqrt(a*x - 1))^2) + integrate(1/2*(a^8*x^8 - 4*a^6*x^6 + 6*a^4* x^4 + (a^4*x^4 + 3)*(a*x + 1)^2*(a*x - 1)^2 + (4*a^5*x^5 - 4*a^3*x^3 + 3*a *x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 4*a^2*x^2 + 3*(2*a^6*x^6 - 4*a^4*x^4 + a^2*x^2 + 1)*(a*x + 1)*(a*x - 1) + (4*a^7*x^7 - 12*a^5*x^5 + 9*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)/((a^8*x^8 + (a*x + 1)^2*(a*x - 1)^ 2*a^4*x^4 - 4*a^6*x^6 + 6*a^4*x^4 + 4*(a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)* (a*x - 1)^(3/2) - 4*a^2*x^2 + 6*(a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*(a*x + 1)* (a*x - 1) + 4*(a^7*x^7 - 3*a^5*x^5 + 3*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a *x - 1) + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)
\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]