Integrand size = 27, antiderivative size = 295 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {16 b^2 d \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {8 b^2 d (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {2 b^2 d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {2 b d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d} \] Output:
-16/75*b^2*d*(-c^2*d*x^2+d)^(1/2)/c^2-8/225*b^2*d*(-c*x+1)*(c*x+1)*(-c^2*d *x^2+d)^(1/2)/c^2-2/125*b^2*d*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2)/c^ 2+2/5*b*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c/(c*x-1)^(1/2)/(c*x+1 )^(1/2)-4/15*b*c*d*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/ 2)/(c*x+1)^(1/2)+2/25*b*c^3*d*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/ (c*x-1)^(1/2)/(c*x+1)^(1/2)-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/ c^2/d
Time = 1.44 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.71 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (225 a^2 \left (-1+c^2 x^2\right )^3-30 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (15-10 c^2 x^2+3 c^4 x^4\right )+2 b^2 \left (-149+187 c^2 x^2-47 c^4 x^4+9 c^6 x^6\right )-30 b \left (-15 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (15-10 c^2 x^2+3 c^4 x^4\right )\right ) \text {arccosh}(c x)+225 b^2 \left (-1+c^2 x^2\right )^3 \text {arccosh}(c x)^2\right )}{1125 c^2 \left (-1+c^2 x^2\right )} \] Input:
Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
-1/1125*(d*Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^3 - 30*a*b*c*x*Sqrt [-1 + c*x]*Sqrt[1 + c*x]*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 2*b^2*(-149 + 187 *c^2*x^2 - 47*c^4*x^4 + 9*c^6*x^6) - 30*b*(-15*a*(-1 + c^2*x^2)^3 + b*c*x* Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcCosh[c*x] + 225*b^2*(-1 + c^2*x^2)^3*ArcCosh[c*x]^2))/(c^2*(-1 + c^2*x^2))
Time = 0.80 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6329, 6304, 6309, 27, 1905, 1576, 1140, 2009}
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.
| \(\displaystyle \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \int (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c^2 x^2-1}}dx}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {c^2 x^2-1}}dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \left (3 \left (c^2 x^2-1\right )^{3/2}-4 \sqrt {c^2 x^2-1}+\frac {8}{\sqrt {c^2 x^2-1}}\right )dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b d \sqrt {d-c^2 d x^2} \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2 d}\) |
Input:
Int[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(c^2*d) + (2*b*d*Sqrt[ d - c^2*d*x^2]*(-1/30*(b*c*Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^2 - (8*(-1 + c^2*x^2)^(3/2))/(3*c^2) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^2)))/( Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + x*(a + b*ArcCosh[c*x]) - (2*c^2*x^3*(a + b *ArcCosh[c*x]))/3 + (c^4*x^5*(a + b*ArcCosh[c*x]))/5))/(5*c*Sqrt[-1 + c*x] *Sqrt[1 + c*x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 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. \(514\) vs. \(2(255)=510\).
Time = 0.77 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.75
| method | result | size |
| orering | \(\frac {\left (549 c^{8} x^{8}-1982 c^{6} x^{6}+4355 c^{4} x^{4}-1420 c^{2} x^{2}+298\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{1125 c^{4} x^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {2 \left (54 c^{6} x^{6}-217 c^{4} x^{4}+672 c^{2} x^{2}-149\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-3 x^{2} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d +\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{1125 c^{4} x^{2} \left (c^{2} x^{2}-1\right )}+\frac {\left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right ) \left (-9 c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}+\frac {4 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}-\frac {12 b \,c^{3} d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{1125 c^{4} x}\) | \(515\) |
| default | \(\text {Expression too large to display}\) | \(1270\) |
| parts | \(\text {Expression too large to display}\) | \(1270\) |
Input:
int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/1125*(549*c^8*x^8-1982*c^6*x^6+4355*c^4*x^4-1420*c^2*x^2+298)/c^4/x^2/(c ^2*x^2-1)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2-2/1125*(54*c^6*x^6-2 17*c^4*x^4+672*c^2*x^2-149)/c^4/x^2/(c^2*x^2-1)*((-c^2*d*x^2+d)^(3/2)*(a+b *arccosh(c*x))^2-3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2*c^2*d+2*x *(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2))+ 1/1125*(9*c^4*x^4-38*c^2*x^2+149)/c^4/x*(-9*c^2*d*x*(-c^2*d*x^2+d)^(1/2)*( a+b*arccosh(c*x))^2+4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))*b*c/(c*x-1)^ (1/2)/(c*x+1)^(1/2)+3*x^3/(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2*c^4*d^ 2-12*b*c^3*d*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c* x+1)^(1/2)+2*x*(-c^2*d*x^2+d)^(3/2)*b^2*c^2/(c*x-1)/(c*x+1)-x*(-c^2*d*x^2+ d)^(3/2)*(a+b*arccosh(c*x))*b*c^2/(c*x-1)^(3/2)/(c*x+1)^(1/2)-x*(-c^2*d*x^ 2+d)^(3/2)*(a+b*arccosh(c*x))*b*c^2/(c*x-1)^(1/2)/(c*x+1)^(3/2))
Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.24 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {225 \, {\left (b^{2} c^{6} d x^{6} - 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} - b^{2} d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (3 \, a b c^{5} d x^{5} - 10 \, a b c^{3} d x^{3} + 15 \, a b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left ({\left (3 \, b^{2} c^{5} d x^{5} - 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 15 \, {\left (a b c^{6} d x^{6} - 3 \, a b c^{4} d x^{4} + 3 \, a b c^{2} d x^{2} - a b d\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} d x^{6} - {\left (675 \, a^{2} + 94 \, b^{2}\right )} c^{4} d x^{4} + {\left (675 \, a^{2} + 374 \, b^{2}\right )} c^{2} d x^{2} - {\left (225 \, a^{2} + 298 \, b^{2}\right )} d\right )} \sqrt {-c^{2} d x^{2} + d}}{1125 \, {\left (c^{4} x^{2} - c^{2}\right )}} \] Input:
integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas ")
Output:
-1/1125*(225*(b^2*c^6*d*x^6 - 3*b^2*c^4*d*x^4 + 3*b^2*c^2*d*x^2 - b^2*d)*s qrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 30*(3*a*b*c^5*d*x^5 - 10*a*b*c^3*d*x^3 + 15*a*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 30*((3*b^2*c^5*d*x^5 - 10*b^2*c^3*d*x^3 + 15*b^2*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 15*(a*b*c^6*d*x^6 - 3*a*b*c^4*d*x^4 + 3*a*b*c^2*d* x^2 - a*b*d)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (9*(25*a ^2 + 2*b^2)*c^6*d*x^6 - (675*a^2 + 94*b^2)*c^4*d*x^4 + (675*a^2 + 374*b^2) *c^2*d*x^2 - (225*a^2 + 298*b^2)*d)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate(x*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**2,x)
Output:
Integral(x*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))**2, x)
Time = 0.06 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.94 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \operatorname {arcosh}\left (c x\right )^{2}}{5 \, c^{2} d} - \frac {2}{1125} \, b^{2} {\left (\frac {9 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{2} x^{4} - 38 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2} x^{2} + \frac {149 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2}}{c^{2}}}{d} - \frac {15 \, {\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} \operatorname {arcosh}\left (c x\right )}{c d}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \operatorname {arcosh}\left (c x\right )}{5 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2}}{5 \, c^{2} d} + \frac {2 \, {\left (3 \, c^{4} \sqrt {-d} d^{2} x^{5} - 10 \, c^{2} \sqrt {-d} d^{2} x^{3} + 15 \, \sqrt {-d} d^{2} x\right )} a b}{75 \, c d} \] Input:
integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima ")
Output:
-1/5*(-c^2*d*x^2 + d)^(5/2)*b^2*arccosh(c*x)^2/(c^2*d) - 2/1125*b^2*((9*sq rt(c^2*x^2 - 1)*c^2*sqrt(-d)*d^2*x^4 - 38*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^2*x ^2 + 149*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^2/c^2)/d - 15*(3*c^4*sqrt(-d)*d^2*x^ 5 - 10*c^2*sqrt(-d)*d^2*x^3 + 15*sqrt(-d)*d^2*x)*arccosh(c*x)/(c*d)) - 2/5 *(-c^2*d*x^2 + d)^(5/2)*a*b*arccosh(c*x)/(c^2*d) - 1/5*(-c^2*d*x^2 + d)^(5 /2)*a^2/(c^2*d) + 2/75*(3*c^4*sqrt(-d)*d^2*x^5 - 10*c^2*sqrt(-d)*d^2*x^3 + 15*sqrt(-d)*d^2*x)*a*b/(c*d)
Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac")
Output:
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. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:
int(x*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(3/2),x)
Output:
int(x*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(3/2), x)
\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, d \left (-\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a^{2}-10 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+10 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x d x \right ) a b \,c^{2}-5 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+5 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}\right )}{5 c^{2}} \] Input:
int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))^2,x)
Output:
(sqrt(d)*d*( - sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 + 2*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt( - c**2*x**2 + 1)*a**2 - 10*int(sqrt( - c**2*x **2 + 1)*acosh(c*x)*x**3,x)*a*b*c**4 + 10*int(sqrt( - c**2*x**2 + 1)*acosh (c*x)*x,x)*a*b*c**2 - 5*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**3,x)*b **2*c**4 + 5*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x,x)*b**2*c**2))/(5* c**2)