Integrand size = 8, antiderivative size = 105 \[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x^2}{3 \text {arccosh}(a x)^2}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)}+\frac {2 \text {Chi}(2 \text {arccosh}(a x))}{3 a^2} \]
1/6/a^2/arccosh(a*x)^2-1/3*x^2/arccosh(a*x)^2+2/3*Chi(2*arccosh(a*x))/a^2- 1/3*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^3-2/3*x*(a*x-1)^(1/2)*(a* x+1)^(1/2)/a/arccosh(a*x)
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25 \[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=\frac {\frac {2 a x-2 a^3 x^3-\sqrt {-1+a x} \sqrt {1+a x} \left (-1+2 a^2 x^2\right ) \text {arccosh}(a x)+\left (4 a x-4 a^3 x^3\right ) \text {arccosh}(a x)^2}{\text {arccosh}(a x)^3}+4 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {Chi}(2 \text {arccosh}(a x))}{6 a^2 \sqrt {-1+a x} \sqrt {1+a x}} \]
((2*a*x - 2*a^3*x^3 - Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-1 + 2*a^2*x^2)*ArcCos h[a*x] + (4*a*x - 4*a^3*x^3)*ArcCosh[a*x]^2)/ArcCosh[a*x]^3 + 4*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*CoshIntegral[2*ArcCosh[a*x]])/(6*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
Time = 1.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6301, 6308, 6366, 6300, 25, 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.
| \(\displaystyle \int \frac {x}{\text {arccosh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {2}{3} a \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx-\frac {\int \frac {1}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx}{3 a}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {2}{3} a \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\int \frac {x}{\text {arccosh}(a x)^2}dx}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}\right )+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {2}{3} a \left (\frac {-\frac {\int -\frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}\right )+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {\int \frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}\right )+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} a \left (-\frac {x^2}{2 a \text {arccosh}(a x)^2}+\frac {-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}+\frac {\int \frac {\sin \left (2 i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}}{a}\right )+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {\text {Chi}(2 \text {arccosh}(a x))}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}\right )+\frac {1}{6 a^2 \text {arccosh}(a x)^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
-1/3*(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) + 1/(6*a^2*ArcCos h[a*x]^2) + (2*a*(-1/2*x^2/(a*ArcCosh[a*x]^2) + (-((x*Sqrt[-1 + a*x]*Sqrt[ 1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[2*ArcCosh[a*x]]/a^2)/a))/3
3.1.68.3.1 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_)^(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[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[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]
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{6 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{6 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{3 \,\operatorname {arccosh}\left (a x \right )}+\frac {2 \,\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{3}}{a^{2}}\) | \(60\) |
| default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{6 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{6 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{3 \,\operatorname {arccosh}\left (a x \right )}+\frac {2 \,\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{3}}{a^{2}}\) | \(60\) |
1/a^2*(-1/6/arccosh(a*x)^3*sinh(2*arccosh(a*x))-1/6/arccosh(a*x)^2*cosh(2* arccosh(a*x))-1/3/arccosh(a*x)*sinh(2*arccosh(a*x))+2/3*Chi(2*arccosh(a*x) ))
\[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=\int \frac {x}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
-1/6*(2*a^12*x^12 - 10*a^10*x^10 + 20*a^8*x^8 - 20*a^6*x^6 + 10*a^4*x^4 + 2*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 2*(5*a^8*x^8 - 9*a ^6*x^6 + 4*a^4*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^9*x^9 - 13*a^7*x^7 + 11*a^5*x^5 - 3*a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a^2*x^2 + 4*(5 *a^10*x^10 - 17*a^8*x^8 + 21*a^6*x^6 - 11*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*( a*x - 1) + (4*a^12*x^12 - 20*a^10*x^10 + 40*a^8*x^8 - 40*a^6*x^6 + 20*a^4* x^4 + 4*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (20*a^8*x^8 - 36*a^6*x^6 + 16*a^4*x^4 + 3*a^2*x^2 - 3)*(a*x + 1)^2*(a*x - 1)^2 + (40*a ^9*x^9 - 104*a^7*x^7 + 88*a^5*x^5 - 21*a^3*x^3 - 3*a*x)*(a*x + 1)^(3/2)*(a *x - 1)^(3/2) - 4*a^2*x^2 + (40*a^10*x^10 - 136*a^8*x^8 + 168*a^6*x^6 - 91 *a^4*x^4 + 22*a^2*x^2 - 3)*(a*x + 1)*(a*x - 1) + (20*a^11*x^11 - 84*a^9*x^ 9 + 136*a^7*x^7 - 107*a^5*x^5 + 42*a^3*x^3 - 7*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*a^11*x^11 - 21*a^9* x^9 + 34*a^7*x^7 - 26*a^5*x^5 + 9*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + (2*a^12*x^12 - 10*a^10*x^10 + 20*a^8*x^8 - 20*a^6*x^6 + 10*a^4*x^4 + 2*(a^7*x^7 - a^5*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (10*a^8*x^8 - 18*a ^6*x^6 + 9*a^4*x^4 - a^2*x^2)*(a*x + 1)^2*(a*x - 1)^2 + (20*a^9*x^9 - 52*a ^7*x^7 + 47*a^5*x^5 - 17*a^3*x^3 + 2*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a^2*x^2 + (20*a^10*x^10 - 68*a^8*x^8 + 87*a^6*x^6 - 51*a^4*x^4 + 13*a^ 2*x^2 - 1)*(a*x + 1)*(a*x - 1) + (10*a^11*x^11 - 42*a^9*x^9 + 69*a^7*x^...
\[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
Timed out. \[ \int \frac {x}{\text {arccosh}(a x)^4} \, dx=\int \frac {x}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]