Integrand size = 6, antiderivative size = 86 \[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}-\frac {x}{6 \text {arccosh}(a x)^2}-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{6 a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{6 a} \]
-1/6*x/arccosh(a*x)^2+1/6*Chi(arccosh(a*x))/a-1/3*(a*x-1)^(1/2)*(a*x+1)^(1 /2)/a/arccosh(a*x)^3-1/6*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)
Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=\frac {\frac {2-2 a^2 x^2-a x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)+\left (1-a^2 x^2\right ) \text {arccosh}(a x)^2}{\text {arccosh}(a x)^3}+\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {Chi}(\text {arccosh}(a x))}{6 a \sqrt {-1+a x} \sqrt {1+a x}} \]
((2 - 2*a^2*x^2 - a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + (1 - a^2 *x^2)*ArcCosh[a*x]^2)/ArcCosh[a*x]^3 + Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x )*CoshIntegral[ArcCosh[a*x]])/(6*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
Time = 0.82 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6295, 6366, 6295, 6368, 3042, 3782}
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.
| \(\displaystyle \int \frac {1}{\text {arccosh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {1}{3} a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\int \frac {1}{\text {arccosh}(a x)^2}dx}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6295 |
| \(\displaystyle \frac {1}{3} a \left (\frac {a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {\int \frac {a x}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}+\frac {1}{3} a \left (-\frac {x}{2 a \text {arccosh}(a x)^2}+\frac {-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}+\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}}{2 a}\right )\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {\text {Chi}(\text {arccosh}(a x))}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
-1/3*(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) + (a*(-1/2*x/(a*Arc Cosh[a*x]^2) + (-((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshI ntegral[ArcCosh[a*x]]/a)/(2*a)))/3
3.1.69.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - 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_)*((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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c *x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{3 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{6 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{6 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{6}}{a}\) | \(67\) |
| default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{3 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{6 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{6 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{6}}{a}\) | \(67\) |
1/a*(-1/3/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/6*a*x/arccosh(a*x)^ 2-1/6/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/6*Chi(arccosh(a*x)))
\[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=\int \frac {1}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
-1/6*(2*a^11*x^11 - 10*a^9*x^9 + 20*a^7*x^7 - 20*a^5*x^5 + 2*(a^6*x^6 - a^ 4*x^4)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 10*a^3*x^3 + 2*(5*a^7*x^7 - 9*a^5 *x^5 + 4*a^3*x^3)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^8*x^8 - 13*a^6*x^6 + 11 *a^4*x^4 - 3*a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^9*x^9 - 17* a^7*x^7 + 21*a^5*x^5 - 11*a^3*x^3 + 2*a*x)*(a*x + 1)*(a*x - 1) + (a^11*x^1 1 - 5*a^9*x^9 + 10*a^7*x^7 - 10*a^5*x^5 + (a^6*x^6 - a^4*x^4 + 3*a^2*x^2 - 3)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 5*a^3*x^3 + (5*a^7*x^7 - 9*a^5*x^5 + 10*a^3*x^3 - 6*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (10*a^8*x^8 - 26*a^6*x^6 + 22*a^4*x^4 - 3*a^2*x^2 - 3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(5*a^9*x^9 - 17*a^7*x^7 + 18*a^5*x^5 - 5*a^3*x^3 - a*x)*(a*x + 1)*(a*x - 1) + (5*a^1 0*x^10 - 21*a^8*x^8 + 31*a^6*x^6 - 20*a^4*x^4 + 6*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))^2 + 2*(5*a^ 10*x^10 - 21*a^8*x^8 + 34*a^6*x^6 - 26*a^4*x^4 + 9*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - 2*a*x + (a^11*x^11 - 5*a^9*x^9 + 10*a^7*x^7 - 10*a^5*x ^5 + (a^6*x^6 - a^2*x^2)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 5*a^3*x^3 + (5* a^7*x^7 - 5*a^5*x^5 - 2*a^3*x^3 + 2*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (10*a^8 *x^8 - 20*a^6*x^6 + 10*a^4*x^4 + a^2*x^2 - 1)*(a*x + 1)^(3/2)*(a*x - 1)^(3 /2) + 2*(5*a^9*x^9 - 15*a^7*x^7 + 16*a^5*x^5 - 7*a^3*x^3 + a*x)*(a*x + 1)* (a*x - 1) + (5*a^10*x^10 - 20*a^8*x^8 + 31*a^6*x^6 - 23*a^4*x^4 + 8*a^2*x^ 2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x)*log(a*x + sqrt(a*x + 1)*sqrt(...
\[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^4} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]