3.5.79 \(\int x^4 (d+e x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\) [479]

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

[Out]

1/5*d^3*x^5*(a+b*arccosh(c*x))+3/7*d^2*e*x^7*(a+b*arccosh(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))+1/1155*b*(231*c^6*d^3+495*c^4*d^2*e+385*c^2*d*e^2+105*e^3)*(-c^2*x^2+1)/c^11/(c*x-1)^(1/2)
/(c*x+1)^(1/2)-1/3465*b*(462*c^6*d^3+1485*c^4*d^2*e+1540*c^2*d*e^2+525*e^3)*(-c^2*x^2+1)^2/c^11/(c*x-1)^(1/2)/
(c*x+1)^(1/2)+1/1925*b*(77*c^6*d^3+495*c^4*d^2*e+770*c^2*d*e^2+350*e^3)*(-c^2*x^2+1)^3/c^11/(c*x-1)^(1/2)/(c*x
+1)^(1/2)-1/1617*b*e*(99*c^4*d^2+308*c^2*d*e+210*e^2)*(-c^2*x^2+1)^4/c^11/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/297*b*
e^2*(11*c^2*d+15*e)*(-c^2*x^2+1)^5/c^11/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/121*b*e^3*(-c^2*x^2+1)^6/c^11/(c*x-1)^(1
/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.44, antiderivative size = 435, normalized size of antiderivative = 1.00, 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 = {276, 5958, 12, 1624, 1813, 1634} \begin {gather*} \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 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}}-\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 (77 c^6 d^3+495 c^4 d^2 e+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 (462 c^6 d^3+1485 c^4 d^2 e+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 (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right )}{1155 c^{11} \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

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 12

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

Rule 276

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 1624

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 1634

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]

Rule 1813

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 5958

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]))

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.25, size = 276, normalized size = 0.63 \begin {gather*} \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 ) \cosh ^{-1}(c x)}{4002075} \end {gather*}

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 = 3.51, size = 335, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{5} d^{3} c^{11} x^{5}+\frac {3}{7} d^{2} c^{11} e \,x^{7}+\frac {1}{3} d \,c^{11} e^{2} x^{9}+\frac {1}{11} e^{3} c^{11} x^{11}\right )}{c^{6}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) d^{3} c^{11} x^{5}}{5}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) d^{2} c^{11} e \,x^{7}}{7}+\frac {\mathrm {arccosh}\left (c x \right ) d \,c^{11} e^{2} x^{9}}{3}+\frac {\mathrm {arccosh}\left (c x \right ) e^{3} c^{11} x^{11}}{11}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 c^{10} e^{3} x^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} e^{3} x^{4}+426888 d^{3} c^{6}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} e^{3} x^{2}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075}\right )}{c^{6}}}{c^{5}}\)	\(335\)
default	\(\frac {\frac {a \left (\frac {1}{5} d^{3} c^{11} x^{5}+\frac {3}{7} d^{2} c^{11} e \,x^{7}+\frac {1}{3} d \,c^{11} e^{2} x^{9}+\frac {1}{11} e^{3} c^{11} x^{11}\right )}{c^{6}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) d^{3} c^{11} x^{5}}{5}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) d^{2} c^{11} e \,x^{7}}{7}+\frac {\mathrm {arccosh}\left (c x \right ) d \,c^{11} e^{2} x^{9}}{3}+\frac {\mathrm {arccosh}\left (c x \right ) e^{3} c^{11} x^{11}}{11}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 c^{10} e^{3} x^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} e^{3} x^{4}+426888 d^{3} c^{6}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} e^{3} x^{2}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075}\right )}{c^{6}}}{c^{5}}\)	\(335\)

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,method=_RETURNVERBOSE)

[Out]

1/c^5*(a/c^6*(1/5*d^3*c^11*x^5+3/7*d^2*c^11*e*x^7+1/3*d*c^11*e^2*x^9+1/11*e^3*c^11*x^11)+b/c^6*(1/5*arccosh(c*
x)*d^3*c^11*x^5+3/7*arccosh(c*x)*d^2*c^11*e*x^7+1/3*arccosh(c*x)*d*c^11*e^2*x^9+1/11*arccosh(c*x)*e^3*c^11*x^1
1-1/4002075*(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 = 0.31, size = 449, normalized size = 1.03 \begin {gather*} \frac {1}{11} \, a x^{11} e^{3} + \frac {1}{3} \, a d x^{9} e^{2} + \frac {3}{7} \, a d^{2} x^{7} e + \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 {gather*}

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*x^11*e^3 + 1/3*a*d*x^9*e^2 + 3/7*a*d^2*x^7*e + 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (381) = 762\).
time = 0.39, size = 822, normalized size = 1.89 \begin {gather*} \frac {363825 \, a c^{11} x^{11} \cosh \left (1\right )^{3} + 363825 \, a c^{11} x^{11} \sinh \left (1\right )^{3} + 1334025 \, a c^{11} d x^{9} \cosh \left (1\right )^{2} + 1715175 \, a c^{11} d^{2} x^{7} \cosh \left (1\right ) + 800415 \, a c^{11} d^{3} x^{5} + 121275 \, {\left (9 \, a c^{11} x^{11} \cosh \left (1\right ) + 11 \, a c^{11} d x^{9}\right )} \sinh \left (1\right )^{2} + 3465 \, {\left (105 \, b c^{11} x^{11} \cosh \left (1\right )^{3} + 105 \, b c^{11} x^{11} \sinh \left (1\right )^{3} + 385 \, b c^{11} d x^{9} \cosh \left (1\right )^{2} + 495 \, b c^{11} d^{2} x^{7} \cosh \left (1\right ) + 231 \, b c^{11} d^{3} x^{5} + 35 \, {\left (9 \, b c^{11} x^{11} \cosh \left (1\right ) + 11 \, b c^{11} d x^{9}\right )} \sinh \left (1\right )^{2} + 5 \, {\left (63 \, b c^{11} x^{11} \cosh \left (1\right )^{2} + 154 \, b c^{11} d x^{9} \cosh \left (1\right ) + 99 \, b c^{11} d^{2} x^{7}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 17325 \, {\left (63 \, a c^{11} x^{11} \cosh \left (1\right )^{2} + 154 \, a c^{11} d x^{9} \cosh \left (1\right ) + 99 \, a c^{11} d^{2} x^{7}\right )} \sinh \left (1\right ) - {\left (160083 \, b c^{10} d^{3} x^{4} + 213444 \, b c^{8} d^{3} x^{2} + 426888 \, b c^{6} d^{3} + 525 \, {\left (63 \, b c^{10} x^{10} + 70 \, b c^{8} x^{8} + 80 \, b c^{6} x^{6} + 96 \, b c^{4} x^{4} + 128 \, b c^{2} x^{2} + 256 \, b\right )} \cosh \left (1\right )^{3} + 525 \, {\left (63 \, b c^{10} x^{10} + 70 \, b c^{8} x^{8} + 80 \, b c^{6} x^{6} + 96 \, b c^{4} x^{4} + 128 \, b c^{2} x^{2} + 256 \, b\right )} \sinh \left (1\right )^{3} + 4235 \, {\left (35 \, b c^{10} d x^{8} + 40 \, b c^{8} d x^{6} + 48 \, b c^{6} d x^{4} + 64 \, b c^{4} d x^{2} + 128 \, b c^{2} d\right )} \cosh \left (1\right )^{2} + 35 \, {\left (4235 \, b c^{10} d x^{8} + 4840 \, b c^{8} d x^{6} + 5808 \, b c^{6} d x^{4} + 7744 \, b c^{4} d x^{2} + 15488 \, b c^{2} d + 45 \, {\left (63 \, b c^{10} x^{10} + 70 \, b c^{8} x^{8} + 80 \, b c^{6} x^{6} + 96 \, b c^{4} x^{4} + 128 \, b c^{2} x^{2} + 256 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 49005 \, {\left (5 \, b c^{10} d^{2} x^{6} + 6 \, b c^{8} d^{2} x^{4} + 8 \, b c^{6} d^{2} x^{2} + 16 \, b c^{4} d^{2}\right )} \cosh \left (1\right ) + 5 \, {\left (49005 \, b c^{10} d^{2} x^{6} + 58806 \, b c^{8} d^{2} x^{4} + 78408 \, b c^{6} d^{2} x^{2} + 156816 \, b c^{4} d^{2} + 315 \, {\left (63 \, b c^{10} x^{10} + 70 \, b c^{8} x^{8} + 80 \, b c^{6} x^{6} + 96 \, b c^{4} x^{4} + 128 \, b c^{2} x^{2} + 256 \, b\right )} \cosh \left (1\right )^{2} + 1694 \, {\left (35 \, b c^{10} d x^{8} + 40 \, b c^{8} d x^{6} + 48 \, b c^{6} d x^{4} + 64 \, b c^{4} d x^{2} + 128 \, b c^{2} d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{4002075 \, c^{11}} \end {gather*}

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*x^11*cosh(1)^3 + 363825*a*c^11*x^11*sinh(1)^3 + 1334025*a*c^11*d*x^9*cosh(1)^2 + 1715
175*a*c^11*d^2*x^7*cosh(1) + 800415*a*c^11*d^3*x^5 + 121275*(9*a*c^11*x^11*cosh(1) + 11*a*c^11*d*x^9)*sinh(1)^
2 + 3465*(105*b*c^11*x^11*cosh(1)^3 + 105*b*c^11*x^11*sinh(1)^3 + 385*b*c^11*d*x^9*cosh(1)^2 + 495*b*c^11*d^2*
x^7*cosh(1) + 231*b*c^11*d^3*x^5 + 35*(9*b*c^11*x^11*cosh(1) + 11*b*c^11*d*x^9)*sinh(1)^2 + 5*(63*b*c^11*x^11*
cosh(1)^2 + 154*b*c^11*d*x^9*cosh(1) + 99*b*c^11*d^2*x^7)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + 17325*(63*a*
c^11*x^11*cosh(1)^2 + 154*a*c^11*d*x^9*cosh(1) + 99*a*c^11*d^2*x^7)*sinh(1) - (160083*b*c^10*d^3*x^4 + 213444*
b*c^8*d^3*x^2 + 426888*b*c^6*d^3 + 525*(63*b*c^10*x^10 + 70*b*c^8*x^8 + 80*b*c^6*x^6 + 96*b*c^4*x^4 + 128*b*c^
2*x^2 + 256*b)*cosh(1)^3 + 525*(63*b*c^10*x^10 + 70*b*c^8*x^8 + 80*b*c^6*x^6 + 96*b*c^4*x^4 + 128*b*c^2*x^2 +
256*b)*sinh(1)^3 + 4235*(35*b*c^10*d*x^8 + 40*b*c^8*d*x^6 + 48*b*c^6*d*x^4 + 64*b*c^4*d*x^2 + 128*b*c^2*d)*cos
h(1)^2 + 35*(4235*b*c^10*d*x^8 + 4840*b*c^8*d*x^6 + 5808*b*c^6*d*x^4 + 7744*b*c^4*d*x^2 + 15488*b*c^2*d + 45*(
63*b*c^10*x^10 + 70*b*c^8*x^8 + 80*b*c^6*x^6 + 96*b*c^4*x^4 + 128*b*c^2*x^2 + 256*b)*cosh(1))*sinh(1)^2 + 4900
5*(5*b*c^10*d^2*x^6 + 6*b*c^8*d^2*x^4 + 8*b*c^6*d^2*x^2 + 16*b*c^4*d^2)*cosh(1) + 5*(49005*b*c^10*d^2*x^6 + 58
806*b*c^8*d^2*x^4 + 78408*b*c^6*d^2*x^2 + 156816*b*c^4*d^2 + 315*(63*b*c^10*x^10 + 70*b*c^8*x^8 + 80*b*c^6*x^6
+ 96*b*c^4*x^4 + 128*b*c^2*x^2 + 256*b)*cosh(1)^2 + 1694*(35*b*c^10*d*x^8 + 40*b*c^8*d*x^6 + 48*b*c^6*d*x^4 +
64*b*c^4*d*x^2 + 128*b*c^2*d)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/c^11

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Sympy [C] Result contains complex when optimal does not.
time = 2.81, size = 638, normalized size = 1.47 \begin {gather*} \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 {gather*}

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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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