Integrand size = 10, antiderivative size = 231 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=-\frac {4144 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^5}-\frac {272 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \text {arccosh}(a x)}{25 a^4}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3 \]
16/25*x*arccosh(a*x)/a^4+8/75*x^3*arccosh(a*x)/a^2+6/125*x^5*arccosh(a*x)+ 1/5*x^5*arccosh(a*x)^3-4144/5625*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5-272/5625* x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-6/625*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/ a-8/25*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5-4/25*x^2*arccosh(a*x )^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-3/25*x^4*arccosh(a*x)^2*(a*x-1)^(1/2)* (a*x+1)^(1/2)/a
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.56 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\frac {-2 \sqrt {-1+a x} \sqrt {1+a x} \left (2072+136 a^2 x^2+27 a^4 x^4\right )+30 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \text {arccosh}(a x)-225 \sqrt {-1+a x} \sqrt {1+a x} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {arccosh}(a x)^2+1125 a^5 x^5 \text {arccosh}(a x)^3}{5625 a^5} \]
(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2072 + 136*a^2*x^2 + 27*a^4*x^4) + 30*a* x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x] - 225*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^2 + 1125*a^5*x^5*ArcCosh[a *x]^3)/(5625*a^5)
Time = 2.18 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.55, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {6298, 6354, 6298, 111, 27, 111, 27, 83, 6354, 6298, 111, 27, 83, 6330, 6294, 83}
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.
| \(\displaystyle \int x^4 \text {arccosh}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \int \frac {x^5 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {2 \int x^4 \text {arccosh}(a x)dx}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \int \frac {x^5}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {\int \frac {4 x^3}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}\right )\right )}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {4 \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}\right )\right )}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}\right )\right )}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}\right )\right )}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}-\frac {2 \int x^2 \text {arccosh}(a x)dx}{3 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{3 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \left (\frac {\int \frac {2 x}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )\right )}{3 a}+\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \left (\frac {2 \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )\right )}{3 a}+\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )\right )}{3 a}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{a^2}-\frac {2 \int \text {arccosh}(a x)dx}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )\right )}{3 a}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{a^2}-\frac {2 \left (x \text {arccosh}(a x)-a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )\right )}{3 a}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {3}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a^2}+\frac {4 \left (\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 a^2}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{a^2}-\frac {2 \left (x \text {arccosh}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a}\right )}{a}\right )}{3 a^2}-\frac {2 \left (\frac {1}{3} x^3 \text {arccosh}(a x)-\frac {1}{3} a \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )\right )}{3 a}\right )}{5 a^2}-\frac {2 \left (\frac {1}{5} x^5 \text {arccosh}(a x)-\frac {1}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a^2}+\frac {4 \left (\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^4}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a^2}\right )}{5 a^2}\right )\right )}{5 a}\right )\) |
(x^5*ArcCosh[a*x]^3)/5 - (3*a*((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a *x]^2)/(5*a^2) - (2*(-1/5*(a*((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a^2) + (4*((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^4) + (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^2)))/(5*a^2))) + (x^5*ArcCosh[a*x])/5))/(5*a) + (4*((x^2*Sqr t[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a^2) - (2*(-1/3*(a*((2*Sqrt[- 1 + a*x]*Sqrt[1 + a*x])/(3*a^4) + (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^ 2))) + (x^3*ArcCosh[a*x])/3))/(3*a) + (2*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Ar cCosh[a*x]^2)/a^2 - (2*(-((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a) + x*ArcCosh[a* x]))/a))/(3*a^2)))/(5*a^2)))/5
3.1.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( -1 + c*x)^p] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
Time = 0.06 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {4 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}+\frac {16 a x \,\operatorname {arccosh}\left (a x \right )}{25}-\frac {4144 \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {6 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )}{125}-\frac {6 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}}{625}-\frac {272 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {8 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{75}}{a^{5}}\) | \(190\) |
| default | \(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {4 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}+\frac {16 a x \,\operatorname {arccosh}\left (a x \right )}{25}-\frac {4144 \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {6 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )}{125}-\frac {6 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}}{625}-\frac {272 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {8 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{75}}{a^{5}}\) | \(190\) |
1/a^5*(1/5*a^5*x^5*arccosh(a*x)^3-8/25*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1 )^(1/2)-3/25*a^4*x^4*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-4/25*a^2*x ^2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+16/25*a*x*arccosh(a*x)-4144/ 5625*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6/125*a^5*x^5*arccosh(a*x)-6/625*(a*x-1)^ (1/2)*(a*x+1)^(1/2)*a^4*x^4-272/5625*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+8 /75*a^3*x^3*arccosh(a*x))
Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\frac {1125 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 225 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 30 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \sqrt {a^{2} x^{2} - 1}}{5625 \, a^{5}} \]
1/5625*(1125*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^3 - 225*(3*a^4*x^4 + 4*a ^2*x^2 + 8)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^2 + 30*(9*a^5*x ^5 + 20*a^3*x^3 + 120*a*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(27*a^4*x^4 + 136*a^2*x^2 + 2072)*sqrt(a^2*x^2 - 1))/a^5
\[ \int x^4 \text {arccosh}(a x)^3 \, dx=\int x^{4} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.71 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcosh}\left (a x\right )^{3} - \frac {1}{25} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} - 1}}{a^{6}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {a^{2} x^{2} - 1} a^{2} x^{4} + 136 \, \sqrt {a^{2} x^{2} - 1} x^{2} + \frac {2072 \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}}{a^{4}} - \frac {15 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \operatorname {arcosh}\left (a x\right )}{a^{5}}\right )} \]
1/5*x^5*arccosh(a*x)^3 - 1/25*(3*sqrt(a^2*x^2 - 1)*x^4/a^2 + 4*sqrt(a^2*x^ 2 - 1)*x^2/a^4 + 8*sqrt(a^2*x^2 - 1)/a^6)*a*arccosh(a*x)^2 - 2/5625*a*((27 *sqrt(a^2*x^2 - 1)*a^2*x^4 + 136*sqrt(a^2*x^2 - 1)*x^2 + 2072*sqrt(a^2*x^2 - 1)/a^2)/a^4 - 15*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*arccosh(a*x)/a^5)
Exception generated. \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\int x^4\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]