3.479 \(\int x^4 (d+e x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=435 \[ \frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^4 \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right )^3 \left (495 c^4 d^2 e+77 c^6 d^3+770 c^2 d e^2+350 e^3\right )}{1925 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left (1-c^2 x^2\right )^2 \left (1485 c^4 d^2 e+462 c^6 d^3+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (495 c^4 d^2 e+231 c^6 d^3+385 c^2 d e^2+105 e^3\right )}{1155 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left (1-c^2 x^2\right )^5 \left (11 c^2 d+15 e\right )}{297 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left (1-c^2 x^2\right )^6}{121 c^{11} \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2*d*e^2 + 105*e^3)*(1 - c^2*x^2))/(1155*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) - (b*(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)^2)/(3465*c^11*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (b*(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)^3)/(1925*c^11*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) - (b*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(1 - c^2*x^2)^4)/(1617*c^11*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (b*e^2*(11*c^2*d + 15*e)*(1 - c^2*x^2)^5)/(297*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(
1 - c^2*x^2)^6)/(121*c^11*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

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Rubi [A]  time = 0.616826, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {270, 5790, 12, 1610, 1799, 1620} \[ \frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^4 \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right )^3 \left (495 c^4 d^2 e+77 c^6 d^3+770 c^2 d e^2+350 e^3\right )}{1925 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left (1-c^2 x^2\right )^2 \left (1485 c^4 d^2 e+462 c^6 d^3+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (495 c^4 d^2 e+231 c^6 d^3+385 c^2 d e^2+105 e^3\right )}{1155 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left (1-c^2 x^2\right )^5 \left (11 c^2 d+15 e\right )}{297 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left (1-c^2 x^2\right )^6}{121 c^{11} \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(b*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2*d*e^2 + 105*e^3)*(1 - c^2*x^2))/(1155*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) - (b*(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)^2)/(3465*c^11*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (b*(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)^3)/(1925*c^11*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) - (b*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(1 - c^2*x^2)^4)/(1617*c^11*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (b*e^2*(11*c^2*d + 15*e)*(1 - c^2*x^2)^5)/(297*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(
1 - c^2*x^2)^6)/(121*c^11*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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5790

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]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[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] && LeQ[m + p, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
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] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[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]

Rule 1620

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]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^4 \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{1155 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{(b c) \int \frac{x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1155}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{1155 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{2310 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3}{c^{10} \sqrt{-1+c^2 x}}+\frac{\left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \sqrt{-1+c^2 x}}{c^{10}}+\frac{3 \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (-1+c^2 x\right )^{3/2}}{c^{10}}+\frac{5 e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (-1+c^2 x\right )^{5/2}}{c^{10}}+\frac{35 e^2 \left (11 c^2 d+15 e\right ) \left (-1+c^2 x\right )^{7/2}}{c^{10}}+\frac{105 e^3 \left (-1+c^2 x\right )^{9/2}}{c^{10}}\right ) \, dx,x,x^2\right )}{2310 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right ) \left (1-c^2 x^2\right )}{1155 c^{11} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \left (1-c^2 x^2\right )^2}{3465 c^{11} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x^2\right )^3}{1925 c^{11} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x^2\right )^4}{1617 c^{11} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^5}{297 c^{11} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b e^3 \left (1-c^2 x^2\right )^6}{121 c^{11} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{5} d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.388668, size = 276, normalized size = 0.63 \[ \frac{3465 a x^5 \left (495 d^2 e x^2+231 d^3+385 d e^2 x^4+105 e^3 x^6\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^{10} x^4 \left (245025 d^2 e x^2+160083 d^3+148225 d e^2 x^4+33075 e^3 x^6\right )+2 c^8 \left (147015 d^2 e x^4+106722 d^3 x^2+84700 d e^2 x^6+18375 e^3 x^8\right )+24 c^6 \left (16335 d^2 e x^2+17787 d^3+8470 d e^2 x^4+1750 e^3 x^6\right )+80 c^4 e \left (9801 d^2+3388 d e x^2+630 e^2 x^4\right )+4480 c^2 e^2 \left (121 d+15 e x^2\right )+134400 e^3\right )}{c^{11}}+3465 b x^5 \cosh ^{-1}(c x) \left (495 d^2 e x^2+231 d^3+385 d e^2 x^4+105 e^3 x^6\right )}{4002075} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(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*Sqrt[-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 +
16335*d^2*e*x^2 + 8470*d*e^2*x^4 + 1750*e^3*x^6) + c^10*x^4*(160083*d^3 + 245025*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])/4002075

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Maple [A]  time = 0.013, size = 335, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{11}{x}^{11}}{11}}+{\frac{d{e}^{2}{c}^{11}{x}^{9}}{3}}+{\frac{3\,{c}^{11}{d}^{2}e{x}^{7}}{7}}+{\frac{{c}^{11}{x}^{5}{d}^{3}}{5}} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{3}{c}^{11}{x}^{11}}{11}}+{\frac{{\rm arccosh} \left (cx\right )d{e}^{2}{c}^{11}{x}^{9}}{3}}+{\frac{3\,{\rm arccosh} \left (cx\right ){c}^{11}{d}^{2}e{x}^{7}}{7}}+{\frac{{\rm arccosh} \left (cx\right ){c}^{11}{x}^{5}{d}^{3}}{5}}-{\frac{33075\,{c}^{10}{e}^{3}{x}^{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\,{d}^{3}{c}^{6}+271040\,{c}^{4}d{e}^{2}{x}^{2}+784080\,{c}^{4}{d}^{2}e+67200\,{c}^{2}{e}^{3}{x}^{2}+542080\,d{e}^{2}{c}^{2}+134400\,{e}^{3}}{4002075}\sqrt{cx-1}\sqrt{cx+1}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x)

[Out]

1/c^5*(a/c^6*(1/11*e^3*c^11*x^11+1/3*d*e^2*c^11*x^9+3/7*c^11*d^2*e*x^7+1/5*c^11*x^5*d^3)+b/c^6*(1/11*arccosh(c
*x)*e^3*c^11*x^11+1/3*arccosh(c*x)*d*e^2*c^11*x^9+3/7*arccosh(c*x)*c^11*d^2*e*x^7+1/5*arccosh(c*x)*c^11*x^5*d^
3-1/4002075*(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+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)))

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Maxima [A]  time = 1.19795, size = 609, normalized size = 1.4 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

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*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*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

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Fricas [A]  time = 2.58607, size = 868, normalized size = 2. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

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 + sq
rt(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 + 78
4080*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*(53
361*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

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Sympy [A]  time = 130.21, size = 638, normalized size = 1.47 \begin{align*} \begin{cases} \frac{a d^{3} x^{5}}{5} + \frac{3 a d^{2} e x^{7}}{7} + \frac{a d e^{2} x^{9}}{3} + \frac{a e^{3} x^{11}}{11} + \frac{b d^{3} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{3 b d^{2} e x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{b d e^{2} x^{9} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{b e^{3} x^{11} \operatorname{acosh}{\left (c x \right )}}{11} - \frac{b d^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{3 b d^{2} e x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{b d e^{2} x^{8} \sqrt{c^{2} x^{2} - 1}}{27 c} - \frac{b e^{3} x^{10} \sqrt{c^{2} x^{2} - 1}}{121 c} - \frac{4 b d^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{3}} - \frac{18 b d^{2} e x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{8 b d e^{2} x^{6} \sqrt{c^{2} x^{2} - 1}}{189 c^{3}} - \frac{10 b e^{3} x^{8} \sqrt{c^{2} x^{2} - 1}}{1089 c^{3}} - \frac{8 b d^{3} \sqrt{c^{2} x^{2} - 1}}{75 c^{5}} - \frac{24 b d^{2} e x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b d e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{315 c^{5}} - \frac{80 b e^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{7623 c^{5}} - \frac{48 b d^{2} e \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} - \frac{64 b d e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{945 c^{7}} - \frac{32 b e^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{2541 c^{7}} - \frac{128 b d e^{2} \sqrt{c^{2} x^{2} - 1}}{945 c^{9}} - \frac{128 b e^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{7623 c^{9}} - \frac{256 b e^{3} \sqrt{c^{2} x^{2} - 1}}{7623 c^{11}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{3} x^{5}}{5} + \frac{3 d^{2} e x^{7}}{7} + \frac{d e^{2} x^{9}}{3} + \frac{e^{3} x^{11}}{11}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)**3*(a+b*acosh(c*x)),x)

[Out]

Piecewise((a*d**3*x**5/5 + 3*a*d**2*e*x**7/7 + a*d*e**2*x**9/3 + a*e**3*x**11/11 + b*d**3*x**5*acosh(c*x)/5 +
3*b*d**2*e*x**7*acosh(c*x)/7 + b*d*e**2*x**9*acosh(c*x)/3 + b*e**3*x**11*acosh(c*x)/11 - b*d**3*x**4*sqrt(c**2
*x**2 - 1)/(25*c) - 3*b*d**2*e*x**6*sqrt(c**2*x**2 - 1)/(49*c) - b*d*e**2*x**8*sqrt(c**2*x**2 - 1)/(27*c) - b*
e**3*x**10*sqrt(c**2*x**2 - 1)/(121*c) - 4*b*d**3*x**2*sqrt(c**2*x**2 - 1)/(75*c**3) - 18*b*d**2*e*x**4*sqrt(c
**2*x**2 - 1)/(245*c**3) - 8*b*d*e**2*x**6*sqrt(c**2*x**2 - 1)/(189*c**3) - 10*b*e**3*x**8*sqrt(c**2*x**2 - 1)
/(1089*c**3) - 8*b*d**3*sqrt(c**2*x**2 - 1)/(75*c**5) - 24*b*d**2*e*x**2*sqrt(c**2*x**2 - 1)/(245*c**5) - 16*b
*d*e**2*x**4*sqrt(c**2*x**2 - 1)/(315*c**5) - 80*b*e**3*x**6*sqrt(c**2*x**2 - 1)/(7623*c**5) - 48*b*d**2*e*sqr
t(c**2*x**2 - 1)/(245*c**7) - 64*b*d*e**2*x**2*sqrt(c**2*x**2 - 1)/(945*c**7) - 32*b*e**3*x**4*sqrt(c**2*x**2
- 1)/(2541*c**7) - 128*b*d*e**2*sqrt(c**2*x**2 - 1)/(945*c**9) - 128*b*e**3*x**2*sqrt(c**2*x**2 - 1)/(7623*c**
9) - 256*b*e**3*sqrt(c**2*x**2 - 1)/(7623*c**11), Ne(c, 0)), ((a + I*pi*b/2)*(d**3*x**5/5 + 3*d**2*e*x**7/7 +
d*e**2*x**9/3 + e**3*x**11/11), True))

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Giac [A]  time = 1.46826, size = 552, normalized size = 1.27 \begin{align*} \frac{1}{5} \, a d^{3} x^{5} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b d^{3} + \frac{1}{7623} \,{\left (693 \, a x^{11} +{\left (693 \, x^{11} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{63 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{11}{2}} + 385 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 990 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 1386 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 693 \, \sqrt{c^{2} x^{2} - 1}}{c^{11}}\right )} b\right )} e^{3} + \frac{1}{945} \,{\left (315 \, a d x^{9} +{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 180 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 420 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} - 1}}{c^{9}}\right )} b d\right )} e^{2} + \frac{3}{245} \,{\left (35 \, a d^{2} x^{7} +{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b d^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

1/5*a*d^3*x^5 + 1/75*(15*x^5*log(c*x + sqrt(c^2*x^2 - 1)) - (3*(c^2*x^2 - 1)^(5/2) + 10*(c^2*x^2 - 1)^(3/2) +
15*sqrt(c^2*x^2 - 1))/c^5)*b*d^3 + 1/7623*(693*a*x^11 + (693*x^11*log(c*x + sqrt(c^2*x^2 - 1)) - (63*(c^2*x^2
- 1)^(11/2) + 385*(c^2*x^2 - 1)^(9/2) + 990*(c^2*x^2 - 1)^(7/2) + 1386*(c^2*x^2 - 1)^(5/2) + 1155*(c^2*x^2 - 1
)^(3/2) + 693*sqrt(c^2*x^2 - 1))/c^11)*b)*e^3 + 1/945*(315*a*d*x^9 + (315*x^9*log(c*x + sqrt(c^2*x^2 - 1)) - (
35*(c^2*x^2 - 1)^(9/2) + 180*(c^2*x^2 - 1)^(7/2) + 378*(c^2*x^2 - 1)^(5/2) + 420*(c^2*x^2 - 1)^(3/2) + 315*sqr
t(c^2*x^2 - 1))/c^9)*b*d)*e^2 + 3/245*(35*a*d^2*x^7 + (35*x^7*log(c*x + sqrt(c^2*x^2 - 1)) - (5*(c^2*x^2 - 1)^
(7/2) + 21*(c^2*x^2 - 1)^(5/2) + 35*(c^2*x^2 - 1)^(3/2) + 35*sqrt(c^2*x^2 - 1))/c^7)*b*d^2)*e