Integrand size = 10, antiderivative size = 155 \[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {x^2}{2 a^2 \text {arccosh}(a x)^2}-\frac {2 x^4}{3 \text {arccosh}(a x)^2}+\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \text {arccosh}(a x)}-\frac {8 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)}+\frac {\text {Chi}(2 \text {arccosh}(a x))}{3 a^4}+\frac {4 \text {Chi}(4 \text {arccosh}(a x))}{3 a^4} \]
1/2*x^2/a^2/arccosh(a*x)^2-2/3*x^4/arccosh(a*x)^2+1/3*Chi(2*arccosh(a*x))/ a^4+4/3*Chi(4*arccosh(a*x))/a^4-1/3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arcc osh(a*x)^3+x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/arccosh(a*x)-8/3*x^3*(a*x-1)^ (1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)
Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=\frac {\sqrt {-1+a x} \left (a x \sqrt {\frac {-1+a x}{1+a x}} \left (2 a^2 x^2-2 a^4 x^4-a x \sqrt {-1+a x} \sqrt {1+a x} \left (-3+4 a^2 x^2\right ) \text {arccosh}(a x)-2 \left (3-11 a^2 x^2+8 a^4 x^4\right ) \text {arccosh}(a x)^2\right )+2 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(2 \text {arccosh}(a x))+8 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(4 \text {arccosh}(a x))\right )}{6 a^4 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^3} \]
(Sqrt[-1 + a*x]*(a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*(2*a^2*x^2 - 2*a^4*x^4 - a *x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-3 + 4*a^2*x^2)*ArcCosh[a*x] - 2*(3 - 11* a^2*x^2 + 8*a^4*x^4)*ArcCosh[a*x]^2) + 2*(-1 + a*x)*ArcCosh[a*x]^3*CoshInt egral[2*ArcCosh[a*x]] + 8*(-1 + a*x)*ArcCosh[a*x]^3*CoshIntegral[4*ArcCosh [a*x]]))/(6*a^4*((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^ 3)
Time = 1.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6301, 6366, 6300, 25, 2009, 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.
| \(\displaystyle \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {4}{3} a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx-\frac {\int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx}{a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle -\frac {\frac {\int \frac {x}{\text {arccosh}(a x)^2}dx}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}}{a}+\frac {4}{3} a \left (\frac {2 \int \frac {x^3}{\text {arccosh}(a x)^2}dx}{a}-\frac {x^4}{2 a \text {arccosh}(a x)^2}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {4}{3} a \left (\frac {2 \left (-\frac {\int \left (-\frac {\cosh (2 \text {arccosh}(a x))}{2 \text {arccosh}(a x)}-\frac {\cosh (4 \text {arccosh}(a x))}{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^4}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{a}-\frac {x^4}{2 a \text {arccosh}(a x)^2}\right )-\frac {\frac {-\frac {\int -\frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}}{a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{3} a \left (\frac {2 \left (-\frac {\int \left (-\frac {\cosh (2 \text {arccosh}(a x))}{2 \text {arccosh}(a x)}-\frac {\cosh (4 \text {arccosh}(a x))}{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^4}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{a}-\frac {x^4}{2 a \text {arccosh}(a x)^2}\right )-\frac {\frac {\frac {\int \frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}}{a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cosh (2 \text {arccosh}(a x))}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}}{a}+\frac {4}{3} a \left (\frac {2 \left (-\frac {-\frac {1}{2} \text {Chi}(2 \text {arccosh}(a x))-\frac {1}{2} \text {Chi}(4 \text {arccosh}(a x))}{a^4}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{a}-\frac {x^4}{2 a \text {arccosh}(a x)^2}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {x^2}{2 a \text {arccosh}(a x)^2}+\frac {-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}+\frac {\int \frac {\sin \left (2 i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a^2}}{a}}{a}+\frac {4}{3} a \left (\frac {2 \left (-\frac {-\frac {1}{2} \text {Chi}(2 \text {arccosh}(a x))-\frac {1}{2} \text {Chi}(4 \text {arccosh}(a x))}{a^4}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{a}-\frac {x^4}{2 a \text {arccosh}(a x)^2}\right )-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {4}{3} a \left (\frac {2 \left (-\frac {-\frac {1}{2} \text {Chi}(2 \text {arccosh}(a x))-\frac {1}{2} \text {Chi}(4 \text {arccosh}(a x))}{a^4}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{a}-\frac {x^4}{2 a \text {arccosh}(a x)^2}\right )-\frac {\frac {\frac {\text {Chi}(2 \text {arccosh}(a x))}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{a}-\frac {x^2}{2 a \text {arccosh}(a x)^2}}{a}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\) |
-1/3*(x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) - (-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)/a + (4*a*(-1/2*x^4/(a*ArcCosh[a*x]^2) + (2*(-((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) - (-1/2*Cosh Integral[2*ArcCosh[a*x]] - CoshIntegral[4*ArcCosh[a*x]]/2)/a^4))/a))/3
3.1.66.3.1 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_)*((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 = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{6 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{3}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{24 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{3 \,\operatorname {arccosh}\left (a x \right )}+\frac {4 \,\operatorname {Chi}\left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{3}}{a^{4}}\) | \(114\) |
| default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{6 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{3}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{24 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{3 \,\operatorname {arccosh}\left (a x \right )}+\frac {4 \,\operatorname {Chi}\left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{3}}{a^{4}}\) | \(114\) |
1/a^4*(-1/12/arccosh(a*x)^3*sinh(2*arccosh(a*x))-1/12/arccosh(a*x)^2*cosh( 2*arccosh(a*x))-1/6/arccosh(a*x)*sinh(2*arccosh(a*x))+1/3*Chi(2*arccosh(a* x))-1/24/arccosh(a*x)^3*sinh(4*arccosh(a*x))-1/12/arccosh(a*x)^2*cosh(4*ar ccosh(a*x))-1/3/arccosh(a*x)*sinh(4*arccosh(a*x))+4/3*Chi(4*arccosh(a*x)))
\[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^{3}}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
-1/6*(2*a^13*x^14 - 10*a^11*x^12 + 20*a^9*x^10 - 20*a^7*x^8 + 10*a^5*x^6 - 2*a^3*x^4 + 2*(a^8*x^9 - a^6*x^7)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 2*(5* a^9*x^10 - 9*a^7*x^8 + 4*a^5*x^6)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x^11 - 13*a^8*x^9 + 11*a^6*x^7 - 3*a^4*x^5)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^11*x^12 - 17*a^9*x^10 + 21*a^7*x^8 - 11*a^5*x^6 + 2*a^3*x^4)*(a*x + 1)*(a*x - 1) + (16*a^13*x^14 - 80*a^11*x^12 + 160*a^9*x^10 - 160*a^7*x^8 + 80*a^5*x^6 - 16*a^3*x^4 + 4*(4*a^8*x^9 - 7*a^6*x^7 + 3*a^4*x^5)*(a*x + 1 )^(5/2)*(a*x - 1)^(5/2) + (80*a^9*x^10 - 192*a^7*x^8 + 154*a^5*x^6 - 45*a^ 3*x^4 + 3*a*x^2)*(a*x + 1)^2*(a*x - 1)^2 + (160*a^10*x^11 - 488*a^8*x^9 + 550*a^6*x^7 - 279*a^4*x^5 + 63*a^2*x^3 - 6*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3 /2) + (160*a^11*x^12 - 592*a^9*x^10 + 846*a^7*x^8 - 583*a^5*x^6 + 196*a^3* x^4 - 27*a*x^2)*(a*x + 1)*(a*x - 1) + (80*a^12*x^13 - 348*a^10*x^11 + 598* a^8*x^9 - 509*a^6*x^7 + 216*a^4*x^5 - 37*a^2*x^3)*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^13 - 21*a^10*x ^11 + 34*a^8*x^9 - 26*a^6*x^7 + 9*a^4*x^5 - a^2*x^3)*sqrt(a*x + 1)*sqrt(a* x - 1) + (4*a^13*x^14 - 20*a^11*x^12 + 40*a^9*x^10 - 40*a^7*x^8 + 20*a^5*x ^6 - 4*a^3*x^4 + 2*(2*a^8*x^9 - 3*a^6*x^7 + a^4*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (20*a^9*x^10 - 44*a^7*x^8 + 31*a^5*x^6 - 7*a^3*x^4)*(a*x + 1) ^2*(a*x - 1)^2 + (40*a^10*x^11 - 116*a^8*x^9 + 121*a^6*x^7 - 53*a^4*x^5 + 8*a^2*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (40*a^11*x^12 - 144*a^9*x^...
Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]