Integrand size = 21, antiderivative size = 341 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))-\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^8 \sqrt {-1+c x} \sqrt {1+c x}} \]
1/4*d^2*x^4*(a+b*arccosh(c*x))+1/3*d*e*x^6*(a+b*arccosh(c*x))+1/8*e^2*x^8* (a+b*arccosh(c*x))+1/3072*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x*(-c^2*x^2+ 1)/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/4608*b*(288*c^4*d^2+320*c^2*d*e+105*e ^2)*x^3*(-c^2*x^2+1)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/1152*b*e*(64*c^2*d+ 21*e)*x^5*(-c^2*x^2+1)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/64*b*e^2*x^7*(-c^ 2*x^2+1)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3072*b*(288*c^4*d^2+320*c^2*d*e+1 05*e^2)*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^8/(c*x-1)^(1/2) /(c*x+1)^(1/2)
Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.63 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \left (315 e^2+30 c^2 e \left (32 d+7 e x^2\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )\right )+384 b c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \text {arccosh}(c x)-6 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{9216 c^8} \]
(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) - b*c*x*Sqrt[-1 + c*x]*Sqrt [1 + c*x]*(315*e^2 + 30*c^2*e*(32*d + 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x ^2 + 21*e^2*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 384*b*c ^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x] - 6*b*(288*c^4*d^2 + 3 20*c^2*d*e + 105*e^2)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(9216*c^8)
Time = 0.56 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6373, 27, 1905, 1590, 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.
\(\displaystyle \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6373 |
\(\displaystyle -b c \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{24} b c \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1905 |
\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 1590 |
\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {x^4 \left (48 c^2 d^2+e \left (64 d c^2+21 e\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 363 |
\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \int \frac {x^4}{\sqrt {c^2 x^2-1}}dx}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \left (\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \left (\frac {3 \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right ) \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\) |
(d^2*x^4*(a + b*ArcCosh[c*x]))/4 + (d*e*x^6*(a + b*ArcCosh[c*x]))/3 + (e^2 *x^8*(a + b*ArcCosh[c*x]))/8 - (b*c*Sqrt[-1 + c^2*x^2]*((3*e^2*x^7*Sqrt[-1 + c^2*x^2])/(8*c^2) + ((e*(64*c^2*d + 21*e)*x^5*Sqrt[-1 + c^2*x^2])/(6*c^ 2) + ((288*c^4*d^2 + 5*e*(64*c^2*d + 21*e))*((x^3*Sqrt[-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*c^2))/(8*c^2)))/(24*Sqrt[-1 + c*x]*Sqrt[1 + c*x] )
3.5.71.3.1 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]}, Sim p[(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] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le Q[m + p, 0]))
Time = 0.76 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.07
method | result | size |
parts | \(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} d^{2} x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccosh}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccosh}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4} d^{2}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (576 c^{7} d^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+512 c^{7} d e \sqrt {c^{2} x^{2}-1}\, x^{5}+144 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+864 c^{5} d^{2} x \sqrt {c^{2} x^{2}-1}+640 \sqrt {c^{2} x^{2}-1}\, c^{5} d e \,x^{3}+168 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+960 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+210 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+960 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+315 e^{2} c x \sqrt {c^{2} x^{2}-1}+315 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 c^{4} \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\) | \(364\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} c^{8} e^{2} x^{8}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (576 c^{7} d^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+512 c^{7} d e \sqrt {c^{2} x^{2}-1}\, x^{5}+144 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+864 c^{5} d^{2} x \sqrt {c^{2} x^{2}-1}+640 \sqrt {c^{2} x^{2}-1}\, c^{5} d e \,x^{3}+168 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+960 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+210 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+960 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+315 e^{2} c x \sqrt {c^{2} x^{2}-1}+315 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}}{c^{4}}\) | \(377\) |
default | \(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} c^{8} e^{2} x^{8}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (576 c^{7} d^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+512 c^{7} d e \sqrt {c^{2} x^{2}-1}\, x^{5}+144 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+864 c^{5} d^{2} x \sqrt {c^{2} x^{2}-1}+640 \sqrt {c^{2} x^{2}-1}\, c^{5} d e \,x^{3}+168 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 c^{4} d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+960 d \,c^{3} e x \sqrt {c^{2} x^{2}-1}+210 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+960 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+315 e^{2} c x \sqrt {c^{2} x^{2}-1}+315 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}}{c^{4}}\) | \(377\) |
a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*d^2*x^4)+b/c^4*(1/8*c^4*arccosh(c*x)*e^2*x^ 8+1/3*c^4*arccosh(c*x)*d*e*x^6+1/4*arccosh(c*x)*c^4*x^4*d^2-1/9216/c^4*(c* x-1)^(1/2)*(c*x+1)^(1/2)*(576*c^7*d^2*(c^2*x^2-1)^(1/2)*x^3+512*c^7*d*e*(c ^2*x^2-1)^(1/2)*x^5+144*e^2*(c^2*x^2-1)^(1/2)*c^7*x^7+864*c^5*d^2*x*(c^2*x ^2-1)^(1/2)+640*(c^2*x^2-1)^(1/2)*c^5*d*e*x^3+168*e^2*c^5*x^5*(c^2*x^2-1)^ (1/2)+864*c^4*d^2*ln(c*x+(c^2*x^2-1)^(1/2))+960*d*c^3*e*x*(c^2*x^2-1)^(1/2 )+210*e^2*(c^2*x^2-1)^(1/2)*c^3*x^3+960*d*c^2*e*ln(c*x+(c^2*x^2-1)^(1/2))+ 315*e^2*c*x*(c^2*x^2-1)^(1/2)+315*e^2*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2- 1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.67 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (144 \, b c^{7} e^{2} x^{7} + 8 \, {\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{8}} \]
1/9216*(1152*a*c^8*e^2*x^8 + 3072*a*c^8*d*e*x^6 + 2304*a*c^8*d^2*x^4 + 3*( 384*b*c^8*e^2*x^8 + 1024*b*c^8*d*e*x^6 + 768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 - 320*b*c^2*d*e - 105*b*e^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (144*b*c^7*e^ 2*x^7 + 8*(64*b*c^7*d*e + 21*b*c^5*e^2)*x^5 + 2*(288*b*c^7*d^2 + 320*b*c^5 *d*e + 105*b*c^3*e^2)*x^3 + 3*(288*b*c^5*d^2 + 320*b*c^3*d*e + 105*b*c*e^2 )*x)*sqrt(c^2*x^2 - 1))/c^8
\[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Time = 0.21 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.97 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d e + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b e^{2} \]
1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 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 + 1/144*(48*x^6*arccosh(c*x) - (8*s qrt(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*d*e + 1/307 2*(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*e^2
Exception generated. \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]