Integrand size = 10, antiderivative size = 170 \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \text {arccosh}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{48 a^5}+\frac {27 \text {Chi}(3 \text {arccosh}(a x))}{32 a^5}+\frac {125 \text {Chi}(5 \text {arccosh}(a x))}{96 a^5} \]
2/3*x^3/a^2/arccosh(a*x)^2-5/6*x^5/arccosh(a*x)^2+1/48*Chi(arccosh(a*x))/a ^5+27/32*Chi(3*arccosh(a*x))/a^5+125/96*Chi(5*arccosh(a*x))/a^5-1/3*x^4*(a *x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^3+2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/ 2)/a^3/arccosh(a*x)-25/6*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)
Leaf count is larger than twice the leaf count of optimal. \(356\) vs. \(2(170)=340\).
Time = 0.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.09 \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\frac {\sqrt {-1+a x} \left (32 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-32 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}}+64 a^3 x^3 \sqrt {-1+a x} \sqrt {\frac {-1+a x}{1+a x}} \sqrt {1+a x} \text {arccosh}(a x)-80 a^5 x^5 \sqrt {-1+a x} \sqrt {\frac {-1+a x}{1+a x}} \sqrt {1+a x} \text {arccosh}(a x)-192 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2+592 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2-400 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2+2 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(\text {arccosh}(a x))+81 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(3 \text {arccosh}(a x))-125 \text {arccosh}(a x)^3 \text {Chi}(5 \text {arccosh}(a x))+125 a x \text {arccosh}(a x)^3 \text {Chi}(5 \text {arccosh}(a x))\right )}{96 a^5 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^3} \]
(Sqrt[-1 + a*x]*(32*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 32*a^6*x^6*Sqrt[( -1 + a*x)/(1 + a*x)] + 64*a^3*x^3*Sqrt[-1 + a*x]*Sqrt[(-1 + a*x)/(1 + a*x) ]*Sqrt[1 + a*x]*ArcCosh[a*x] - 80*a^5*x^5*Sqrt[-1 + a*x]*Sqrt[(-1 + a*x)/( 1 + a*x)]*Sqrt[1 + a*x]*ArcCosh[a*x] - 192*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a* x)]*ArcCosh[a*x]^2 + 592*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 - 400*a^6*x^6*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 + 2*(-1 + a*x)*Ar cCosh[a*x]^3*CoshIntegral[ArcCosh[a*x]] + 81*(-1 + a*x)*ArcCosh[a*x]^3*Cos hIntegral[3*ArcCosh[a*x]] - 125*ArcCosh[a*x]^3*CoshIntegral[5*ArcCosh[a*x] ] + 125*a*x*ArcCosh[a*x]^3*CoshIntegral[5*ArcCosh[a*x]]))/(96*a^5*((-1 + a *x)/(1 + a*x))^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^3)
Time = 1.04 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6301, 6366, 6300, 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.
| \(\displaystyle \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {5}{3} a \int \frac {x^5}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx-\frac {4 \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx}{3 a}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {5}{3} a \left (\frac {5 \int \frac {x^4}{\text {arccosh}(a x)^2}dx}{2 a}-\frac {x^5}{2 a \text {arccosh}(a x)^2}\right )-\frac {4 \left (\frac {3 \int \frac {x^2}{\text {arccosh}(a x)^2}dx}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {5}{3} a \left (\frac {5 \left (-\frac {\int \left (-\frac {a x}{8 \text {arccosh}(a x)}-\frac {9 \cosh (3 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}-\frac {5 \cosh (5 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^5}{2 a \text {arccosh}(a x)^2}\right )-\frac {4 \left (\frac {3 \left (-\frac {\int \left (-\frac {a x}{4 \text {arccosh}(a x)}-\frac {3 \cosh (3 \text {arccosh}(a x))}{4 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{3} a \left (\frac {5 \left (-\frac {-\frac {1}{8} \text {Chi}(\text {arccosh}(a x))-\frac {9}{16} \text {Chi}(3 \text {arccosh}(a x))-\frac {5}{16} \text {Chi}(5 \text {arccosh}(a x))}{a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^5}{2 a \text {arccosh}(a x)^2}\right )-\frac {4 \left (\frac {3 \left (-\frac {-\frac {1}{4} \text {Chi}(\text {arccosh}(a x))-\frac {3}{4} \text {Chi}(3 \text {arccosh}(a x))}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
-1/3*(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) - (4*(-1/2*x^3/ (a*ArcCosh[a*x]^2) + (3*(-((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a *x])) - (-1/4*CoshIntegral[ArcCosh[a*x]] - (3*CoshIntegral[3*ArcCosh[a*x]] )/4)/a^3))/(2*a)))/(3*a) + (5*a*(-1/2*x^5/(a*ArcCosh[a*x]^2) + (5*(-((x^4* Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) - (-1/8*CoshIntegral[ArcCo sh[a*x]] - (9*CoshIntegral[3*ArcCosh[a*x]])/16 - (5*CoshIntegral[5*ArcCosh [a*x]])/16)/a^5))/(2*a)))/3
3.1.65.3.1 Defintions of rubi rules used
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_)*((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 = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{48 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{48 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{48}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {3 \cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {9 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \,\operatorname {arccosh}\left (a x \right )}+\frac {27 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{48 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {5 \cosh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {25 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \,\operatorname {arccosh}\left (a x \right )}+\frac {125 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96}}{a^{5}}\) | \(175\) |
| default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{48 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{48 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{48}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {3 \cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {9 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \,\operatorname {arccosh}\left (a x \right )}+\frac {27 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{48 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {5 \cosh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {25 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \,\operatorname {arccosh}\left (a x \right )}+\frac {125 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96}}{a^{5}}\) | \(175\) |
1/a^5*(-1/24/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/48*a*x/arccosh(a *x)^2-1/48/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/48*Chi(arccosh(a*x)) -1/16/arccosh(a*x)^3*sinh(3*arccosh(a*x))-3/32/arccosh(a*x)^2*cosh(3*arcco sh(a*x))-9/32/arccosh(a*x)*sinh(3*arccosh(a*x))+27/32*Chi(3*arccosh(a*x))- 1/48/arccosh(a*x)^3*sinh(5*arccosh(a*x))-5/96/arccosh(a*x)^2*cosh(5*arccos h(a*x))-25/96/arccosh(a*x)*sinh(5*arccosh(a*x))+125/96*Chi(5*arccosh(a*x)) )
\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^{4}}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
-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*x + 1)^(5/2)*(a*x - 1)^(5/2) + 2*(5 *a^9*x^11 - 9*a^7*x^9 + 4*a^5*x^7)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x^1 2 - 13*a^8*x^10 + 11*a^6*x^8 - 3*a^4*x^6)*(a*x + 1)^(3/2)*(a*x - 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*x + 1)*(a*x - 1) + (25*a^13*x^15 - 125*a^11*x^13 + 250*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*x + 1)^(5/2)*(a*x - 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*x + 1)^2*(a*x - 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 *x + 1)^(3/2)*(a*x - 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*x + 1)*(a*x - 1) + (125*a^1 2*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*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^ 2 + 2*(5*a^12*x^14 - 21*a^10*x^12 + 34*a^8*x^10 - 26*a^6*x^8 + 9*a^4*x^6 - a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x - 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^10 - 8*a^6*x^8 + 3*a^4*x^6)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (25*a^9*x^11 - 57*a^7*x^9 + 4 2*a^5*x^7 - 10*a^3*x^5)*(a*x + 1)^2*(a*x - 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*x + 1)^(3/2)*(a*x - 1)...
\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]