Integrand size = 21, antiderivative size = 365 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right ) \sqrt {-1+c x} \sqrt {1+c x}}{1155 c^{11}}-\frac {b \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{3465 c^{11}}-\frac {b \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) (-1+c x)^{5/2} (1+c x)^{5/2}}{1925 c^{11}}-\frac {b e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) (-1+c x)^{7/2} (1+c x)^{7/2}}{1617 c^{11}}-\frac {b e^2 \left (11 c^2 d+15 e\right ) (-1+c x)^{9/2} (1+c x)^{9/2}}{297 c^{11}}-\frac {b e^3 (-1+c x)^{11/2} (1+c x)^{11/2}}{121 c^{11}}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x)) \] Output:
-1/1155*b*(231*c^6*d^3+495*c^4*d^2*e+385*c^2*d*e^2+105*e^3)*(c*x-1)^(1/2)* (c*x+1)^(1/2)/c^11-1/3465*b*(462*c^6*d^3+1485*c^4*d^2*e+1540*c^2*d*e^2+525 *e^3)*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^11-1/1925*b*(77*c^6*d^3+495*c^4*d^2*e+ 770*c^2*d*e^2+350*e^3)*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^11-1/1617*b*e*(99*c^4 *d^2+308*c^2*d*e+210*e^2)*(c*x-1)^(7/2)*(c*x+1)^(7/2)/c^11-1/297*b*e^2*(11 *c^2*d+15*e)*(c*x-1)^(9/2)*(c*x+1)^(9/2)/c^11-1/121*b*e^3*(c*x-1)^(11/2)*( c*x+1)^(11/2)/c^11+1/5*d^3*x^5*(a+b*arccosh(c*x))+3/7*d^2*e*x^7*(a+b*arcco sh(c*x))+1/3*d*e^2*x^9*(a+b*arccosh(c*x))+1/11*e^3*x^11*(a+b*arccosh(c*x))
Time = 0.27 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.76 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {3465 a x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (134400 e^3+4480 c^2 e^2 \left (121 d+15 e x^2\right )+80 c^4 e \left (9801 d^2+3388 d e x^2+630 e^2 x^4\right )+24 c^6 \left (17787 d^3+16335 d^2 e x^2+8470 d e^2 x^4+1750 e^3 x^6\right )+c^{10} x^4 \left (160083 d^3+245025 d^2 e x^2+148225 d e^2 x^4+33075 e^3 x^6\right )+2 c^8 \left (106722 d^3 x^2+147015 d^2 e x^4+84700 d e^2 x^6+18375 e^3 x^8\right )\right )}{c^{11}}+3465 b x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right ) \text {arccosh}(c x)}{4002075} \] Input:
Integrate[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
Output:
(3465*a*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) - (b*S qrt[-1 + c*x]*Sqrt[1 + c*x]*(134400*e^3 + 4480*c^2*e^2*(121*d + 15*e*x^2) + 80*c^4*e*(9801*d^2 + 3388*d*e*x^2 + 630*e^2*x^4) + 24*c^6*(17787*d^3 + 1 6335*d^2*e*x^2 + 8470*d*e^2*x^4 + 1750*e^3*x^6) + c^10*x^4*(160083*d^3 + 2 45025*d^2*e*x^2 + 148225*d*e^2*x^4 + 33075*e^3*x^6) + 2*c^8*(106722*d^3*x^ 2 + 147015*d^2*e*x^4 + 84700*d*e^2*x^6 + 18375*e^3*x^8)))/c^11 + 3465*b*x^ 5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6)*ArcCosh[c*x])/40 02075
Time = 0.98 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6373, 27, 2113, 2331, 2123, 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.
| \(\displaystyle \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle -b c \int \frac {x^5 \left (105 e^3 x^6+385 d e^2 x^4+495 d^2 e x^2+231 d^3\right )}{1155 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {x^5 \left (105 e^3 x^6+385 d e^2 x^4+495 d^2 e x^2+231 d^3\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{1155}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2113 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^5 \left (105 e^3 x^6+385 d e^2 x^4+495 d^2 e x^2+231 d^3\right )}{\sqrt {c^2 x^2-1}}dx}{1155 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (105 e^3 x^6+385 d e^2 x^4+495 d^2 e x^2+231 d^3\right )}{\sqrt {c^2 x^2-1}}dx^2}{2310 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {105 e^3 \left (c^2 x^2-1\right )^{9/2}}{c^{10}}+\frac {35 e^2 \left (11 d c^2+15 e\right ) \left (c^2 x^2-1\right )^{7/2}}{c^{10}}+\frac {5 e \left (99 d^2 c^4+308 d e c^2+210 e^2\right ) \left (c^2 x^2-1\right )^{5/2}}{c^{10}}+\frac {3 \left (77 d^3 c^6+495 d^2 e c^4+770 d e^2 c^2+350 e^3\right ) \left (c^2 x^2-1\right )^{3/2}}{c^{10}}+\frac {\left (462 d^3 c^6+1485 d^2 e c^4+1540 d e^2 c^2+525 e^3\right ) \sqrt {c^2 x^2-1}}{c^{10}}+\frac {231 d^3 c^6+495 d^2 e c^4+385 d e^2 c^2+105 e^3}{c^{10} \sqrt {c^2 x^2-1}}\right )dx^2}{2310 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 e^2 \left (c^2 x^2-1\right )^{9/2} \left (11 c^2 d+15 e\right )}{9 c^{12}}+\frac {210 e^3 \left (c^2 x^2-1\right )^{11/2}}{11 c^{12}}+\frac {10 e \left (c^2 x^2-1\right )^{7/2} \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{7 c^{12}}+\frac {6 \left (c^2 x^2-1\right )^{5/2} \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right )}{5 c^{12}}+\frac {2 \left (c^2 x^2-1\right )^{3/2} \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right )}{3 c^{12}}+\frac {2 \sqrt {c^2 x^2-1} \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right )}{c^{12}}\right )}{2310 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
Output:
-1/2310*(b*c*Sqrt[-1 + c^2*x^2]*((2*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2 *d*e^2 + 105*e^3)*Sqrt[-1 + c^2*x^2])/c^12 + (2*(462*c^6*d^3 + 1485*c^4*d^ 2*e + 1540*c^2*d*e^2 + 525*e^3)*(-1 + c^2*x^2)^(3/2))/(3*c^12) + (6*(77*c^ 6*d^3 + 495*c^4*d^2*e + 770*c^2*d*e^2 + 350*e^3)*(-1 + c^2*x^2)^(5/2))/(5* c^12) + (10*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(-1 + c^2*x^2)^(7/2))/( 7*c^12) + (70*e^2*(11*c^2*d + 15*e)*(-1 + c^2*x^2)^(9/2))/(9*c^12) + (210* e^3*(-1 + c^2*x^2)^(11/2))/(11*c^12)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d ^3*x^5*(a + b*ArcCosh[c*x]))/5 + (3*d^2*e*x^7*(a + b*ArcCosh[c*x]))/7 + (d *e^2*x^9*(a + b*ArcCosh[c*x]))/3 + (e^3*x^11*(a + b*ArcCosh[c*x]))/11
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. )*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ m]/(a*c + b*d*x^2)^FracPart[m]) Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a *d, 0] && EqQ[m, n] && !IntegerQ[m]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
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 = 0.26 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.87
| method | result | size |
| parts | \(a \left (\frac {1}{11} e^{3} x^{11}+\frac {1}{3} d \,e^{2} x^{9}+\frac {3}{7} d^{2} e \,x^{7}+\frac {1}{5} d^{3} x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e^{3} x^{11}}{11}+\frac {c^{5} \operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{9}}{3}+\frac {3 c^{5} \operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d^{3}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 e^{3} x^{10} c^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} e^{3} x^{4}+426888 c^{6} d^{3}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} e^{3} x^{2}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075 c^{6}}\right )}{c^{5}}\) | \(319\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} c^{11} d^{3} x^{5}+\frac {3}{7} c^{11} d^{2} e \,x^{7}+\frac {1}{3} c^{11} d \,e^{2} x^{9}+\frac {1}{11} e^{3} c^{11} x^{11}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d^{3} x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{11} d^{2} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d \,e^{2} x^{9}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{11} x^{11}}{11}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 e^{3} x^{10} c^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} e^{3} x^{4}+426888 c^{6} d^{3}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} e^{3} x^{2}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075}\right )}{c^{6}}}{c^{5}}\) | \(335\) |
| default | \(\frac {\frac {a \left (\frac {1}{5} c^{11} d^{3} x^{5}+\frac {3}{7} c^{11} d^{2} e \,x^{7}+\frac {1}{3} c^{11} d \,e^{2} x^{9}+\frac {1}{11} e^{3} c^{11} x^{11}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d^{3} x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{11} d^{2} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d \,e^{2} x^{9}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{11} x^{11}}{11}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 e^{3} x^{10} c^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} e^{3} x^{4}+426888 c^{6} d^{3}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} e^{3} x^{2}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075}\right )}{c^{6}}}{c^{5}}\) | \(335\) |
| orering | \(\frac {\left (694575 c^{12} e^{4} x^{14}+3312400 c^{12} d \,e^{3} x^{12}+6092350 c^{12} d^{2} e^{2} x^{10}+36750 c^{10} e^{4} x^{12}+5096520 c^{12} d^{3} e \,x^{8}+226450 c^{10} d \,e^{3} x^{10}+1440747 c^{12} d^{4} x^{6}+574750 c^{10} d^{2} e^{2} x^{8}+52500 c^{8} e^{4} x^{10}+729630 c^{10} d^{3} e \,x^{6}+359800 c^{8} d \,e^{3} x^{8}+213444 c^{10} d^{4} x^{4}+1115620 c^{8} d^{2} e^{2} x^{6}+84000 c^{6} e^{4} x^{8}+2526480 c^{8} d^{3} e \,x^{4}+711200 c^{6} d \,e^{3} x^{6}+853776 c^{8} d^{4} x^{2}+4191440 c^{6} d^{2} e^{2} x^{4}+168000 c^{4} e^{4} x^{6}-2700720 c^{6} d^{3} e \,x^{2}+2777600 c^{4} d \,e^{3} x^{4}-1707552 c^{6} d^{4}-6756640 c^{4} d^{2} e^{2} x^{2}+672000 c^{2} e^{4} x^{4}-3136320 c^{4} d^{3} e -5152000 c^{2} d \,e^{3} x^{2}-2168320 c^{2} d^{2} e^{2}-1344000 e^{4} x^{2}-537600 d \,e^{3}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{4002075 x \left (e \,x^{2}+d \right ) c^{12}}-\frac {\left (33075 e^{3} x^{10} c^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} e^{3} x^{4}+426888 c^{6} d^{3}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} e^{3} x^{2}+542080 c^{2} d \,e^{2}+134400 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (4 x^{3} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+6 x^{5} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e +\frac {x^{4} \left (e \,x^{2}+d \right )^{3} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{4002075 c^{12} \left (e \,x^{2}+d \right )^{3} x^{4}}\) | \(654\) |
Input:
int(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/11*e^3*x^11+1/3*d*e^2*x^9+3/7*d^2*e*x^7+1/5*d^3*x^5)+b/c^5*(1/11*c^5* arccosh(c*x)*e^3*x^11+1/3*c^5*arccosh(c*x)*d*e^2*x^9+3/7*c^5*arccosh(c*x)* d^2*e*x^7+1/5*arccosh(c*x)*c^5*x^5*d^3-1/4002075/c^6*(c*x-1)^(1/2)*(c*x+1) ^(1/2)*(33075*c^10*e^3*x^10+148225*c^10*d*e^2*x^8+245025*c^10*d^2*e*x^6+36 750*c^8*e^3*x^8+160083*c^10*d^3*x^4+169400*c^8*d*e^2*x^6+294030*c^8*d^2*e* x^4+42000*c^6*e^3*x^6+213444*c^8*d^3*x^2+203280*c^6*d*e^2*x^4+392040*c^6*d ^2*e*x^2+50400*c^4*e^3*x^4+426888*c^6*d^3+271040*c^4*d*e^2*x^2+784080*c^4* d^2*e+67200*c^2*e^3*x^2+542080*c^2*d*e^2+134400*e^3))
Time = 0.10 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.92 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {363825 \, a c^{11} e^{3} x^{11} + 1334025 \, a c^{11} d e^{2} x^{9} + 1715175 \, a c^{11} d^{2} e x^{7} + 800415 \, a c^{11} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} e^{3} x^{11} + 385 \, b c^{11} d e^{2} x^{9} + 495 \, b c^{11} d^{2} e x^{7} + 231 \, b c^{11} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (33075 \, b c^{10} e^{3} x^{10} + 426888 \, b c^{6} d^{3} + 1225 \, {\left (121 \, b c^{10} d e^{2} + 30 \, b c^{8} e^{3}\right )} x^{8} + 784080 \, b c^{4} d^{2} e + 25 \, {\left (9801 \, b c^{10} d^{2} e + 6776 \, b c^{8} d e^{2} + 1680 \, b c^{6} e^{3}\right )} x^{6} + 542080 \, b c^{2} d e^{2} + 3 \, {\left (53361 \, b c^{10} d^{3} + 98010 \, b c^{8} d^{2} e + 67760 \, b c^{6} d e^{2} + 16800 \, b c^{4} e^{3}\right )} x^{4} + 134400 \, b e^{3} + 4 \, {\left (53361 \, b c^{8} d^{3} + 98010 \, b c^{6} d^{2} e + 67760 \, b c^{4} d e^{2} + 16800 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{4002075 \, c^{11}} \] Input:
integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
1/4002075*(363825*a*c^11*e^3*x^11 + 1334025*a*c^11*d*e^2*x^9 + 1715175*a*c ^11*d^2*e*x^7 + 800415*a*c^11*d^3*x^5 + 3465*(105*b*c^11*e^3*x^11 + 385*b* c^11*d*e^2*x^9 + 495*b*c^11*d^2*e*x^7 + 231*b*c^11*d^3*x^5)*log(c*x + sqrt (c^2*x^2 - 1)) - (33075*b*c^10*e^3*x^10 + 426888*b*c^6*d^3 + 1225*(121*b*c ^10*d*e^2 + 30*b*c^8*e^3)*x^8 + 784080*b*c^4*d^2*e + 25*(9801*b*c^10*d^2*e + 6776*b*c^8*d*e^2 + 1680*b*c^6*e^3)*x^6 + 542080*b*c^2*d*e^2 + 3*(53361* b*c^10*d^3 + 98010*b*c^8*d^2*e + 67760*b*c^6*d*e^2 + 16800*b*c^4*e^3)*x^4 + 134400*b*e^3 + 4*(53361*b*c^8*d^3 + 98010*b*c^6*d^2*e + 67760*b*c^4*d*e^ 2 + 16800*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))/c^11
Timed out. \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:
integrate(x**4*(e*x**2+d)**3*(a+b*acosh(c*x)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.24 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{11} \, a e^{3} x^{11} + \frac {1}{3} \, a d e^{2} x^{9} + \frac {3}{7} \, a d^{2} e x^{7} + \frac {1}{5} \, a d^{3} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{3} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d^{2} e + \frac {1}{945} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b d e^{2} + \frac {1}{7623} \, {\left (693 \, x^{11} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {63 \, \sqrt {c^{2} x^{2} - 1} x^{10}}{c^{2}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{6}} + \frac {96 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{10}} + \frac {256 \, \sqrt {c^{2} x^{2} - 1}}{c^{12}}\right )} c\right )} b e^{3} \] Input:
integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
1/11*a*e^3*x^11 + 1/3*a*d*e^2*x^9 + 3/7*a*d^2*e*x^7 + 1/5*a*d^3*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^3 + 3/245*(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^2*e + 1/945*(315*x^9*a rccosh(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*d*e^2 + 1/7623*(693*x^11*arccosh(c*x) - (63*sqrt(c ^2*x^2 - 1)*x^10/c^2 + 70*sqrt(c^2*x^2 - 1)*x^8/c^4 + 80*sqrt(c^2*x^2 - 1) *x^6/c^6 + 96*sqrt(c^2*x^2 - 1)*x^4/c^8 + 128*sqrt(c^2*x^2 - 1)*x^2/c^10 + 256*sqrt(c^2*x^2 - 1)/c^12)*c)*b*e^3
Exception generated. \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")
Output:
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. \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int(x^4*(a + b*acosh(c*x))*(d + e*x^2)^3,x)
Output:
int(x^4*(a + b*acosh(c*x))*(d + e*x^2)^3, x)
Time = 0.21 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.40 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {-134400 \sqrt {c^{2} x^{2}-1}\, b \,e^{3}-36750 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} e^{3} x^{8}-42000 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e^{3} x^{6}-784080 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d^{2} e -50400 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{3} x^{4}-542080 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d \,e^{2}-67200 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{3} x^{2}+1715175 a \,c^{11} d^{2} e \,x^{7}+1334025 a \,c^{11} d \,e^{2} x^{9}+1715175 \mathit {acosh} \left (c x \right ) b \,c^{11} d^{2} e \,x^{7}+1334025 \mathit {acosh} \left (c x \right ) b \,c^{11} d \,e^{2} x^{9}-245025 \sqrt {c^{2} x^{2}-1}\, b \,c^{10} d^{2} e \,x^{6}-148225 \sqrt {c^{2} x^{2}-1}\, b \,c^{10} d \,e^{2} x^{8}-294030 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{2} e \,x^{4}-169400 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d \,e^{2} x^{6}-392040 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{2} e \,x^{2}-203280 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d \,e^{2} x^{4}-271040 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d \,e^{2} x^{2}-426888 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{3}+800415 a \,c^{11} d^{3} x^{5}+363825 a \,c^{11} e^{3} x^{11}+800415 \mathit {acosh} \left (c x \right ) b \,c^{11} d^{3} x^{5}+363825 \mathit {acosh} \left (c x \right ) b \,c^{11} e^{3} x^{11}-160083 \sqrt {c^{2} x^{2}-1}\, b \,c^{10} d^{3} x^{4}-33075 \sqrt {c^{2} x^{2}-1}\, b \,c^{10} e^{3} x^{10}-213444 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} d^{3} x^{2}}{4002075 c^{11}} \] Input:
int(x^4*(e*x^2+d)^3*(a+b*acosh(c*x)),x)
Output:
(800415*acosh(c*x)*b*c**11*d**3*x**5 + 1715175*acosh(c*x)*b*c**11*d**2*e*x **7 + 1334025*acosh(c*x)*b*c**11*d*e**2*x**9 + 363825*acosh(c*x)*b*c**11*e **3*x**11 - 160083*sqrt(c**2*x**2 - 1)*b*c**10*d**3*x**4 - 245025*sqrt(c** 2*x**2 - 1)*b*c**10*d**2*e*x**6 - 148225*sqrt(c**2*x**2 - 1)*b*c**10*d*e** 2*x**8 - 33075*sqrt(c**2*x**2 - 1)*b*c**10*e**3*x**10 - 213444*sqrt(c**2*x **2 - 1)*b*c**8*d**3*x**2 - 294030*sqrt(c**2*x**2 - 1)*b*c**8*d**2*e*x**4 - 169400*sqrt(c**2*x**2 - 1)*b*c**8*d*e**2*x**6 - 36750*sqrt(c**2*x**2 - 1 )*b*c**8*e**3*x**8 - 426888*sqrt(c**2*x**2 - 1)*b*c**6*d**3 - 392040*sqrt( c**2*x**2 - 1)*b*c**6*d**2*e*x**2 - 203280*sqrt(c**2*x**2 - 1)*b*c**6*d*e* *2*x**4 - 42000*sqrt(c**2*x**2 - 1)*b*c**6*e**3*x**6 - 784080*sqrt(c**2*x* *2 - 1)*b*c**4*d**2*e - 271040*sqrt(c**2*x**2 - 1)*b*c**4*d*e**2*x**2 - 50 400*sqrt(c**2*x**2 - 1)*b*c**4*e**3*x**4 - 542080*sqrt(c**2*x**2 - 1)*b*c* *2*d*e**2 - 67200*sqrt(c**2*x**2 - 1)*b*c**2*e**3*x**2 - 134400*sqrt(c**2* x**2 - 1)*b*e**3 + 800415*a*c**11*d**3*x**5 + 1715175*a*c**11*d**2*e*x**7 + 1334025*a*c**11*d*e**2*x**9 + 363825*a*c**11*e**3*x**11)/(4002075*c**11)