Integrand size = 10, antiderivative size = 183 \[ \int x^3 \text {arccosh}(a x)^3 \, dx=-\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {45 \text {arccosh}(a x)}{256 a^4}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}-\frac {3 \text {arccosh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^3 \]
-45/256*arccosh(a*x)/a^4+9/32*x^2*arccosh(a*x)/a^2+3/32*x^4*arccosh(a*x)-3 /32*arccosh(a*x)^3/a^4+1/4*x^4*arccosh(a*x)^3-45/256*x*(a*x-1)^(1/2)*(a*x+ 1)^(1/2)/a^3-3/128*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-9/32*x*arccosh(a*x)^2 *(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-3/16*x^3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a* x+1)^(1/2)/a
Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.78 \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\frac {-3 a x \sqrt {-1+a x} \sqrt {1+a x} \left (15+2 a^2 x^2\right )+24 a^2 x^2 \left (3+a^2 x^2\right ) \text {arccosh}(a x)-24 a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right ) \text {arccosh}(a x)^2+8 \left (-3+8 a^4 x^4\right ) \text {arccosh}(a x)^3-45 \log \left (a x+\sqrt {-1+a x} \sqrt {1+a x}\right )}{256 a^4} \]
(-3*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(15 + 2*a^2*x^2) + 24*a^2*x^2*(3 + a^ 2*x^2)*ArcCosh[a*x] - 24*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(3 + 2*a^2*x^2)* ArcCosh[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcCosh[a*x]^3 - 45*Log[a*x + Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/(256*a^4)
Time = 1.75 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.45, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6298, 6354, 6298, 111, 27, 101, 43, 6354, 6298, 101, 43, 6308}
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.
| \(\displaystyle \int x^3 \text {arccosh}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {\int x^3 \text {arccosh}(a x)dx}{2 a}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \int \frac {x^4}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {\int \frac {3 x^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}\right )}{2 a}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {3 \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}\right )}{2 a}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}\right )\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}\right )}{2 a}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}\right )\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )}{2 a}\right )\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int x \text {arccosh}(a x)dx}{a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )}{2 a}\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {1}{2} a \int \frac {x^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )}{2 a}\right )\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \left (-\frac {\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{a}+\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )}{2 a}\right )\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 a^2}-\frac {\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {1}{2} a \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{a}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )}{2 a}\right )\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)^3}{6 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 a^2}-\frac {\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {1}{2} a \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{a}\right )}{4 a^2}-\frac {\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )}{2 a}\right )\) |
(x^4*ArcCosh[a*x]^3)/4 - (3*a*((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a *x]^2)/(4*a^2) - ((x^4*ArcCosh[a*x])/4 - (a*((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*a^2) + (3*((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a^2) + ArcCosh[a*x ]/(2*a^3)))/(4*a^2)))/4)/(2*a) + (3*((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCo sh[a*x]^2)/(2*a^2) + ArcCosh[a*x]^3/(6*a^3) - ((x^2*ArcCosh[a*x])/2 - (a*( (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a^2) + ArcCosh[a*x]/(2*a^3)))/2)/a))/( 4*a^2)))/4
3.1.23.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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_.)*((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_.)/(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_)*((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 = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {9 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{32}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{128}-\frac {45 \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{256}-\frac {45 \,\operatorname {arccosh}\left (a x \right )}{256}+\frac {9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{32}}{a^{4}}\) | \(150\) |
| default | \(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {9 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{32}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{128}-\frac {45 \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{256}-\frac {45 \,\operatorname {arccosh}\left (a x \right )}{256}+\frac {9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{32}}{a^{4}}\) | \(150\) |
1/a^4*(1/4*a^4*x^4*arccosh(a*x)^3-3/16*a^3*x^3*arccosh(a*x)^2*(a*x-1)^(1/2 )*(a*x+1)^(1/2)-9/32*a*x*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3/32*a rccosh(a*x)^3+3/32*a^4*x^4*arccosh(a*x)-3/128*a^3*x^3*(a*x-1)^(1/2)*(a*x+1 )^(1/2)-45/256*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-45/256*arccosh(a*x)+9/32*a^ 2*x^2*arccosh(a*x))
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.78 \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 24 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{256 \, a^{4}} \]
1/256*(8*(8*a^4*x^4 - 3)*log(a*x + sqrt(a^2*x^2 - 1))^3 - 24*(2*a^3*x^3 + 3*a*x)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^2 + 3*(8*a^4*x^4 + 2 4*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 - 1)) - 3*(2*a^3*x^3 + 15*a*x)*sqrt (a^2*x^2 - 1))/a^4
\[ \int x^3 \text {arccosh}(a x)^3 \, dx=\int x^{3} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]
\[ \int x^3 \text {arccosh}(a x)^3 \, dx=\int { x^{3} \operatorname {arcosh}\left (a x\right )^{3} \,d x } \]
1/4*x^4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3 - integrate(3/4*(a^3*x^6 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^5 - a*x^4)*log(a*x + sqrt(a*x + 1)*sqr t(a*x - 1))^2/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x), x)
Exception generated. \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\int x^3\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]