Integrand size = 25, antiderivative size = 200 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {73 b d^2 x \sqrt {-1+c x} \sqrt {1+c x}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4608 c}+\frac {43 b c d^2 x^5 \sqrt {-1+c x} \sqrt {1+c x}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {-1+c x} \sqrt {1+c x}-\frac {73 b d^2 \text {arccosh}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x)) \]
-73/3072*b*d^2*arccosh(c*x)/c^4+1/4*d^2*x^4*(a+b*arccosh(c*x))-1/3*c^2*d^2 *x^6*(a+b*arccosh(c*x))+1/8*c^4*d^2*x^8*(a+b*arccosh(c*x))-73/3072*b*d^2*x *(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-73/4608*b*d^2*x^3*(c*x-1)^(1/2)*(c*x+1)^( 1/2)/c+43/1152*b*c*d^2*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/64*b*c^3*d^2*x^7* (c*x-1)^(1/2)*(c*x+1)^(1/2)
Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (2304 a c^4 x^4-3072 a c^6 x^6+1152 a c^8 x^8-219 b c x \sqrt {-1+c x} \sqrt {1+c x}-146 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+344 b c^5 x^5 \sqrt {-1+c x} \sqrt {1+c x}-144 b c^7 x^7 \sqrt {-1+c x} \sqrt {1+c x}+384 b c^4 x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)-438 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{9216 c^4} \]
(d^2*(2304*a*c^4*x^4 - 3072*a*c^6*x^6 + 1152*a*c^8*x^8 - 219*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 146*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 344*b *c^5*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 144*b*c^7*x^7*Sqrt[-1 + c*x]*Sqrt[ 1 + c*x] + 384*b*c^4*x^4*(6 - 8*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] - 438*b* ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(9216*c^4)
Time = 0.52 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6336, 27, 1905, 1590, 27, 363, 262, 262, 224, 219}
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.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle -b c \int \frac {d^2 x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{24} b c d^2 \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {\int \frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {c^2 x^2-1}}dx}{8 c^2}+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{8} \int \frac {x^4 \left (48-43 c^2 x^2\right )}{\sqrt {c^2 x^2-1}}dx+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {c^2 x^2-1}}dx-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{8} c^4 d^2 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 d^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))-\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\) |
(d^2*x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*d^2*x^6*(a + b*ArcCosh[c*x]))/3 + (c^4*d^2*x^8*(a + b*ArcCosh[c*x]))/8 - (b*c*d^2*Sqrt[-1 + c^2*x^2]*((3*c^2 *x^7*Sqrt[-1 + c^2*x^2])/8 + ((-43*x^5*Sqrt[-1 + c^2*x^2])/6 + (73*((x^3*S qrt[-1 + c^2*x^2])/(4*c^2) + (3*((x*Sqrt[-1 + c^2*x^2])/(2*c^2) + ArcTanh[ (c*x)/Sqrt[-1 + c^2*x^2]]/(2*c^3)))/(4*c^2)))/6)/8))/(24*Sqrt[-1 + c*x]*Sq rt[1 + c*x])
3.1.11.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && E qQ[c^2*d + e, 0] && IGtQ[p, 0]
Time = 0.51 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96
| method | result | size |
| parts | \(d^{2} a \left (\frac {1}{8} c^{4} x^{8}-\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (144 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-344 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+146 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+219 c x \sqrt {c^{2} x^{2}-1}+219 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(192\) |
| derivativedivides | \(\frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (144 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-344 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+146 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+219 c x \sqrt {c^{2} x^{2}-1}+219 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(196\) |
| default | \(\frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (144 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-344 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+146 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+219 c x \sqrt {c^{2} x^{2}-1}+219 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(196\) |
d^2*a*(1/8*c^4*x^8-1/3*c^2*x^6+1/4*x^4)+d^2*b/c^4*(1/8*arccosh(c*x)*c^8*x^ 8-1/3*arccosh(c*x)*c^6*x^6+1/4*c^4*x^4*arccosh(c*x)-1/9216*(c*x-1)^(1/2)*( c*x+1)^(1/2)*(144*c^7*x^7*(c^2*x^2-1)^(1/2)-344*c^5*x^5*(c^2*x^2-1)^(1/2)+ 146*(c^2*x^2-1)^(1/2)*c^3*x^3+219*c*x*(c^2*x^2-1)^(1/2)+219*ln(c*x+(c^2*x^ 2-1)^(1/2)))/(c^2*x^2-1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.80 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1152 \, a c^{8} d^{2} x^{8} - 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} - 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (144 \, b c^{7} d^{2} x^{7} - 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} + 219 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{4}} \]
1/9216*(1152*a*c^8*d^2*x^8 - 3072*a*c^6*d^2*x^6 + 2304*a*c^4*d^2*x^4 + 3*( 384*b*c^8*d^2*x^8 - 1024*b*c^6*d^2*x^6 + 768*b*c^4*d^2*x^4 - 73*b*d^2)*log (c*x + sqrt(c^2*x^2 - 1)) - (144*b*c^7*d^2*x^7 - 344*b*c^5*d^2*x^5 + 146*b *c^3*d^2*x^3 + 219*b*c*d^2*x)*sqrt(c^2*x^2 - 1))/c^4
\[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=d^{2} \left (\int a x^{3}\, dx + \int \left (- 2 a c^{2} x^{5}\right )\, dx + \int a c^{4} x^{7}\, dx + \int b x^{3} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 b c^{2} x^{5} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{7} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a*x**3, x) + Integral(-2*a*c**2*x**5, x) + Integral(a*c**4* x**7, x) + Integral(b*x**3*acosh(c*x), x) + Integral(-2*b*c**2*x**5*acosh( c*x), x) + Integral(b*c**4*x**7*acosh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (168) = 336\).
Time = 0.25 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.73 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \, a c^{4} d^{2} x^{8} - \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} - \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} \]
1/8*a*c^4*d^2*x^8 - 1/3*a*c^2*d^2*x^6 + 1/3072*(384*x^8*arccosh(c*x) - (48 *sqrt(c^2*x^2 - 1)*x^7/c^2 + 56*sqrt(c^2*x^2 - 1)*x^5/c^4 + 70*sqrt(c^2*x^ 2 - 1)*x^3/c^6 + 105*sqrt(c^2*x^2 - 1)*x/c^8 + 105*log(2*c^2*x + 2*sqrt(c^ 2*x^2 - 1)*c)/c^9)*c)*b*c^4*d^2 + 1/4*a*d^2*x^4 - 1/144*(48*x^6*arccosh(c* x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt (c^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*c^ 2*d^2 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c ^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*d^2
Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]