Integrand size = 21, antiderivative size = 319 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x)) \]
1/5*d^2*x^5*(a+b*arccosh(c*x))+2/7*d*e*x^7*(a+b*arccosh(c*x))+1/9*e^2*x^9* (a+b*arccosh(c*x))+1/315*b*(63*c^4*d^2+90*c^2*d*e+35*e^2)*(-c^2*x^2+1)/c^9 /(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/945*b*(63*c^4*d^2+135*c^2*d*e+70*e^2)*(-c^2 *x^2+1)^2/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/525*b*(21*c^4*d^2+90*c^2*d*e+7 0*e^2)*(-c^2*x^2+1)^3/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/441*b*e*(9*c^2*d+1 4*e)*(-c^2*x^2+1)^4/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/81*b*e^2*(-c^2*x^2+1 )^5/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.60 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^2+160 c^2 e \left (81 d+14 e x^2\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )\right )}{c^9}+315 b x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right ) \text {arccosh}(c x)}{99225} \]
(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^2 + 160*c^2*e*(81*d + 14*e*x^2) + 24*c^4*(441*d^2 + 270*d*e* x^2 + 70*e^2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) + c^ 8*(3969*d^2*x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcCosh[c*x])/99225
Time = 0.60 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6373, 27, 1905, 1578, 1195, 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.
| \(\displaystyle \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {35 e^2 \left (c^2 x^2-1\right )^{7/2}}{c^8}+\frac {10 e \left (9 d c^2+14 e\right ) \left (c^2 x^2-1\right )^{5/2}}{c^8}+\frac {3 \left (21 d^2 c^4+90 d e c^2+70 e^2\right ) \left (c^2 x^2-1\right )^{3/2}}{c^8}+\frac {2 \left (63 d^2 c^4+135 d e c^2+70 e^2\right ) \sqrt {c^2 x^2-1}}{c^8}+\frac {63 d^2 c^4+90 d e c^2+35 e^2}{c^8 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))+\frac {2}{7} d e x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {20 e \left (c^2 x^2-1\right )^{7/2} \left (9 c^2 d+14 e\right )}{7 c^{10}}+\frac {70 e^2 \left (c^2 x^2-1\right )^{9/2}}{9 c^{10}}+\frac {6 \left (c^2 x^2-1\right )^{5/2} \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{5 c^{10}}+\frac {4 \left (c^2 x^2-1\right )^{3/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{3 c^{10}}+\frac {2 \sqrt {c^2 x^2-1} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{c^{10}}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\) |
-1/630*(b*c*Sqrt[-1 + c^2*x^2]*((2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*Sqrt [-1 + c^2*x^2])/c^10 + (4*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(-1 + c^2*x^ 2)^(3/2))/(3*c^10) + (6*(21*c^4*d^2 + 90*c^2*d*e + 70*e^2)*(-1 + c^2*x^2)^ (5/2))/(5*c^10) + (20*e*(9*c^2*d + 14*e)*(-1 + c^2*x^2)^(7/2))/(7*c^10) + (70*e^2*(-1 + c^2*x^2)^(9/2))/(9*c^10)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*x^5*(a + b*ArcCosh[c*x]))/5 + (2*d*e*x^7*(a + b*ArcCosh[c*x]))/7 + (e ^2*x^9*(a + b*ArcCosh[c*x]))/9
3.5.70.3.1 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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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 egerQ[(m - 1)/2]
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 = 0.84 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.67
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{2} x^{9}+\frac {2}{7} d e \,x^{7}+\frac {1}{5} d^{2} x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e^{2} x^{9}}{9}+\frac {2 c^{5} \operatorname {arccosh}\left (c x \right ) d e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d^{2}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{2} x^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 c^{6} e^{2} x^{6}+4860 c^{6} d e \,x^{4}+5292 c^{6} d^{2} x^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 e^{2} c^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225 c^{4}}\right )}{c^{5}}\) | \(214\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} c^{9} d^{2} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} x^{5}}{5}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{2} x^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 c^{6} e^{2} x^{6}+4860 c^{6} d e \,x^{4}+5292 c^{6} d^{2} x^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 e^{2} c^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225}\right )}{c^{4}}}{c^{5}}\) | \(227\) |
| default | \(\frac {\frac {a \left (\frac {1}{5} c^{9} d^{2} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} d^{2} x^{5}}{5}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{2} x^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 c^{6} e^{2} x^{6}+4860 c^{6} d e \,x^{4}+5292 c^{6} d^{2} x^{2}+1680 c^{4} e^{2} x^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 e^{2} c^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right )}{99225}\right )}{c^{4}}}{c^{5}}\) | \(227\) |
a*(1/9*e^2*x^9+2/7*d*e*x^7+1/5*d^2*x^5)+b/c^5*(1/9*c^5*arccosh(c*x)*e^2*x^ 9+2/7*c^5*arccosh(c*x)*d*e*x^7+1/5*arccosh(c*x)*c^5*x^5*d^2-1/99225/c^4*(c *x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*e^2*x^8+4050*c^8*d*e*x^6+3969*c^8*d^2* x^4+1400*c^6*e^2*x^6+4860*c^6*d*e*x^4+5292*c^6*d^2*x^2+1680*c^4*e^2*x^4+64 80*c^4*d*e*x^2+10584*c^4*d^2+2240*c^2*e^2*x^2+12960*c^2*d*e+4480*e^2))
Time = 0.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.72 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} e^{2} x^{9} + 28350 \, a c^{9} d e x^{7} + 19845 \, a c^{9} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} e^{2} x^{9} + 90 \, b c^{9} d e x^{7} + 63 \, b c^{9} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} e^{2} x^{8} + 10584 \, b c^{4} d^{2} + 50 \, {\left (81 \, b c^{8} d e + 28 \, b c^{6} e^{2}\right )} x^{6} + 12960 \, b c^{2} d e + 3 \, {\left (1323 \, b c^{8} d^{2} + 1620 \, b c^{6} d e + 560 \, b c^{4} e^{2}\right )} x^{4} + 4480 \, b e^{2} + 4 \, {\left (1323 \, b c^{6} d^{2} + 1620 \, b c^{4} d e + 560 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \]
1/99225*(11025*a*c^9*e^2*x^9 + 28350*a*c^9*d*e*x^7 + 19845*a*c^9*d^2*x^5 + 315*(35*b*c^9*e^2*x^9 + 90*b*c^9*d*e*x^7 + 63*b*c^9*d^2*x^5)*log(c*x + sq rt(c^2*x^2 - 1)) - (1225*b*c^8*e^2*x^8 + 10584*b*c^4*d^2 + 50*(81*b*c^8*d* e + 28*b*c^6*e^2)*x^6 + 12960*b*c^2*d*e + 3*(1323*b*c^8*d^2 + 1620*b*c^6*d *e + 560*b*c^4*e^2)*x^4 + 4480*b*e^2 + 4*(1323*b*c^6*d^2 + 1620*b*c^4*d*e + 560*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^9
\[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \]
Time = 0.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.96 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{2} \]
1/9*a*e^2*x^9 + 2/7*a*d*e*x^7 + 1/5*a*d^2*x^5 + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2* x^2 - 1)/c^6)*c)*b*d^2 + 2/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1) *x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16* sqrt(c^2*x^2 - 1)/c^8)*c)*b*d*e + 1/2835*(315*x^9*arccosh(c*x) - (35*sqrt( c^2*x^2 - 1)*x^8/c^2 + 40*sqrt(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x^2 - 1) *x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c^10)*c)*b *e^2
Exception generated. \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^4 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]