Integrand size = 27, antiderivative size = 186 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {14 b^2 \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {2}{27} b^2 x^2 \sqrt {d-c^2 d x^2}+\frac {2 b x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2/c^2/d-14/27*b^2*(-c^2*d*x^2 +d)^(1/2)/c^2+2/27*b^2*x^2*(-c^2*d*x^2+d)^(1/2)+2/3*b*x*(a+b*arccosh(c*x)) *(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/9*b*c*x^3*(a+b*arcco sh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.58 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.97 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-6 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-3+c^2 x^2\right )+9 a^2 \left (-1+c^2 x^2\right )^2+2 b^2 \left (7-8 c^2 x^2+c^4 x^4\right )+6 b \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (3-c^2 x^2\right )+3 a \left (-1+c^2 x^2\right )^2\right ) \text {arccosh}(c x)+9 b^2 \left (-1+c^2 x^2\right )^2 \text {arccosh}(c x)^2\right )}{27 c^2 \left (-1+c^2 x^2\right )} \]
(Sqrt[d - c^2*d*x^2]*(-6*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-3 + c^2*x^ 2) + 9*a^2*(-1 + c^2*x^2)^2 + 2*b^2*(7 - 8*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x *Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - c^2*x^2) + 3*a*(-1 + c^2*x^2)^2)*ArcCos h[c*x] + 9*b^2*(-1 + c^2*x^2)^2*ArcCosh[c*x]^2))/(27*c^2*(-1 + c^2*x^2))
Time = 0.57 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6329, 25, 6304, 6309, 27, 960, 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.
| \(\displaystyle \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle -\frac {2 b \sqrt {d-c^2 d x^2} \int -((1-c x) (c x+1) (a+b \text {arccosh}(c x)))dx}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \left (\frac {7}{3} \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(c^2*d) + (2*b*Sqrt[d - c^2*d*x^2]*(-1/3*(b*c*((7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2) - (x^2*S qrt[-1 + c*x]*Sqrt[1 + c*x])/3)) + x*(a + b*ArcCosh[c*x]) - (c^2*x^3*(a + b*ArcCosh[c*x]))/3))/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.2.72.3.1 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[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n *(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ (b1*b2*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n /2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] , x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(725\) vs. \(2(158)=316\).
Time = 0.56 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.90
| method | result | size |
| default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}-6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}+6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\) | \(726\) |
| parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}-6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}+6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\) | \(726\) |
-1/3*a^2*(-c^2*d*x^2+d)^(3/2)/c^2/d+b^2*(1/216*(-d*(c^2*x^2-1))^(1/2)*(4*c ^4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-3*(c*x-1)^(1/2)*(c* x+1)^(1/2)*c*x+1)*(9*arccosh(c*x)^2-6*arccosh(c*x)+2)/(c*x+1)/c^2/(c*x-1)- 1/8*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(ar ccosh(c*x)^2-2*arccosh(c*x)+2)/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1 /2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2+2*arccosh (c*x)+2)/(c*x+1)/c^2/(c*x-1)+1/216*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2 )*(c*x+1)^(1/2)*c^3*x^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2* x^2+1)*(9*arccosh(c*x)^2+6*arccosh(c*x)+2)/(c*x+1)/c^2/(c*x-1))+2*a*b*(1/7 2*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2 )*c^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(-1+3*arccosh(c*x))/(c*x+1) /c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c ^2*x^2-1)*(-1+arccosh(c*x))/(c*x+1)/c^2/(c*x-1)-1/8*(-d*(c^2*x^2-1))^(1/2) *(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(1+arccosh(c*x))/(c*x+1)/c^2 /(c*x-1)+1/72*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x ^3+4*c^4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*(1+3*arccosh(c *x))/(c*x+1)/c^2/(c*x-1))
Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.51 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {9 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a b c^{3} x^{3} - 3 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left ({\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 3 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - 2 \, {\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]
1/27*(9*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(a*b*c^3*x^3 - 3*a*b*c*x)*sqrt(-c^2*d*x^2 + d)*s qrt(c^2*x^2 - 1) - 6*((b^2*c^3*x^3 - 3*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt( c^2*x^2 - 1) - 3*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*sqrt(-c^2*d*x^2 + d)) *log(c*x + sqrt(c^2*x^2 - 1)) + ((9*a^2 + 2*b^2)*c^4*x^4 - 2*(9*a^2 + 8*b^ 2)*c^2*x^2 + 9*a^2 + 14*b^2)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
\[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
Time = 0.34 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.10 \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-d} d x^{2} - \frac {7 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d}{c^{2}}}{d} - \frac {3 \, {\left (c^{2} \sqrt {-d} d x^{3} - 3 \, \sqrt {-d} d x\right )} \operatorname {arcosh}\left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{3 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \operatorname {arcosh}\left (c x\right )}{3 \, c^{2} d} - \frac {2 \, {\left (c^{2} \sqrt {-d} d x^{3} - 3 \, \sqrt {-d} d x\right )} a b}{9 \, c d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d} \]
2/27*b^2*((sqrt(c^2*x^2 - 1)*sqrt(-d)*d*x^2 - 7*sqrt(c^2*x^2 - 1)*sqrt(-d) *d/c^2)/d - 3*(c^2*sqrt(-d)*d*x^3 - 3*sqrt(-d)*d*x)*arccosh(c*x)/(c*d)) - 1/3*(-c^2*d*x^2 + d)^(3/2)*b^2*arccosh(c*x)^2/(c^2*d) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a*b*arccosh(c*x)/(c^2*d) - 2/9*(c^2*sqrt(-d)*d*x^3 - 3*sqrt(-d)*d *x)*a*b/(c*d) - 1/3*(-c^2*d*x^2 + d)^(3/2)*a^2/(c^2*d)
Exception generated. \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]