Integrand size = 29, antiderivative size = 355 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt {d-c^2 d x^2}}+\frac {15 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{64 c^5 \sqrt {d-c^2 d x^2}}-\frac {3 b x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}} \]
-15/64*b^2*x*(-c*x+1)*(c*x+1)/c^4/(-c^2*d*x^2+d)^(1/2)-1/32*b^2*x^3*(-c*x+ 1)*(c*x+1)/c^2/(-c^2*d*x^2+d)^(1/2)+15/64*b^2*arccosh(c*x)*(c*x-1)^(1/2)*( c*x+1)^(1/2)/c^5/(-c^2*d*x^2+d)^(1/2)-3/8*b*x^2*(a+b*arccosh(c*x))*(c*x-1) ^(1/2)*(c*x+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)-1/8*b*x^4*(a+b*arccosh(c*x)) *(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/8*(a+b*arccosh(c*x)) ^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c^5/(-c^2*d*x^2+d)^(1/2)-3/8*x*(a+b*arcco sh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d-1/4*x^3*(a+b*arccosh(c*x))^2*(-c^2*d *x^2+d)^(1/2)/c^2/d
Time = 1.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {32 a^2 c \sqrt {d} x \left (-1+c^2 x^2\right ) \left (3+2 c^2 x^2\right )-96 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (32 \text {arccosh}(c x)^3-4 \text {arccosh}(c x) (16 \cosh (2 \text {arccosh}(c x))+\cosh (4 \text {arccosh}(c x)))+32 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))+8 \text {arccosh}(c x)^2 (8 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x)))\right )-4 a b \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (16 \cosh (2 \text {arccosh}(c x))+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) (6 \text {arccosh}(c x)+8 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))))}{256 c^5 \sqrt {d} \sqrt {d-c^2 d x^2}} \]
(32*a^2*c*Sqrt[d]*x*(-1 + c^2*x^2)*(3 + 2*c^2*x^2) - 96*a^2*Sqrt[d - c^2*d *x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b^2*Sqr t[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(32*ArcCosh[c*x]^3 - 4*ArcCosh[c *x]*(16*Cosh[2*ArcCosh[c*x]] + Cosh[4*ArcCosh[c*x]]) + 32*Sinh[2*ArcCosh[c *x]] + Sinh[4*ArcCosh[c*x]] + 8*ArcCosh[c*x]^2*(8*Sinh[2*ArcCosh[c*x]] + S inh[4*ArcCosh[c*x]])) - 4*a*b*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) *(16*Cosh[2*ArcCosh[c*x]] + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*(6*ArcCo sh[c*x] + 8*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]])))/(256*c^5*Sqrt[d ]*Sqrt[d - c^2*d*x^2])
Time = 1.41 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {6353, 6298, 111, 27, 101, 43, 6353, 6298, 101, 43, 6307}
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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int x^3 (a+b \text {arccosh}(c x))dx}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle \frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \int x (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {3 \left (-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6307 |
| \(\displaystyle -\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 c^2 d}+\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 c^2 d}+\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{4 c^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\) |
-1/4*(x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((x^4*(a + b*ArcCosh[c*x]))/4 - (b*c*((x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c^2) + (3*((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2 ) + ArcCosh[c*x]/(2*c^3)))/(4*c^2)))/4))/(2*c*Sqrt[d - c^2*d*x^2]) + (3*(- 1/2*(x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) + (Sqrt[-1 + c* x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]) - ( b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((x^2*(a + b*ArcCosh[c*x]))/2 - (b*c*((x*Sq rt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/2))/(c*Sqrt[d - c^2*d*x^2])))/(4*c^2)
3.2.95.3.1 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[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x ] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 *p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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*Ar cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1091\) vs. \(2(307)=614\).
Time = 0.93 (sec) , antiderivative size = 1092, normalized size of antiderivative = 3.08
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1092\) |
| parts | \(\text {Expression too large to display}\) | \(1092\) |
-1/4*a^2*x^3/c^2/d*(-c^2*d*x^2+d)^(1/2)-3/8*a^2/c^4*x/d*(-c^2*d*x^2+d)^(1/ 2)+3/8*a^2/c^4/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+ b^2*(-1/8*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^5/(c^2*x^ 2-1)*arccosh(c*x)^3-1/512*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*( c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2 *x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*arccosh(c*x)^2-4*arccosh(c*x)+1)/d/c^ 5/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2) *(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(2*arccosh(c*x)^2-2*ar ccosh(c*x)+1)/d/c^5/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1 /2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(2* arccosh(c*x)^2+2*arccosh(c*x)+1)/d/c^5/(c^2*x^2-1)-1/512*(-d*(c^2*x^2-1))^ (1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(1/2)*(c *x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(8*arcco sh(c*x)^2+4*arccosh(c*x)+1)/d/c^5/(c^2*x^2-1))+2*a*b*(-3/16*(-d*(c^2*x^2-1 ))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^5/(c^2*x^2-1)*arccosh(c*x)^2-1/25 6*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/ 2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))*(-1+4*arccosh(c*x))/d/c^5/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2 )*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c* x+1)^(1/2))*(-1+2*arccosh(c*x))/d/c^5/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))...
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
integral(-(b^2*x^4*arccosh(c*x)^2 + 2*a*b*x^4*arccosh(c*x) + a^2*x^4)*sqrt (-c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
-1/8*a^2*(2*sqrt(-c^2*d*x^2 + d)*x^3/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*x/(c ^4*d) - 3*arcsin(c*x)/(c^5*sqrt(d))) + integrate(b^2*x^4*log(c*x + sqrt(c* x + 1)*sqrt(c*x - 1))^2/sqrt(-c^2*d*x^2 + d) + 2*a*b*x^4*log(c*x + sqrt(c* x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]