3.5.88 \(\int (d+e x^2)^4 (a+b \cosh ^{-1}(c x)) \, dx\) [488]

Optimal. Leaf size=395 \[ \frac {b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right ) \]

[Out]

d^4*x*(a+b*arccosh(c*x))+4/3*d^3*e*x^3*(a+b*arccosh(c*x))+6/5*d^2*e^2*x^5*(a+b*arccosh(c*x))+4/7*d*e^3*x^7*(a+
b*arccosh(c*x))+1/9*e^4*x^9*(a+b*arccosh(c*x))+1/315*b*(315*c^8*d^4+420*c^6*d^3*e+378*c^4*d^2*e^2+180*c^2*d*e^
3+35*e^4)*(-c^2*x^2+1)/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/945*b*e*(105*c^6*d^3+189*c^4*d^2*e+135*c^2*d*e^2+35*e
^3)*(-c^2*x^2+1)^2/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/525*b*e^2*(63*c^4*d^2+90*c^2*d*e+35*e^2)*(-c^2*x^2+1)^3/c
^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/441*b*e^3*(9*c^2*d+7*e)*(-c^2*x^2+1)^4/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/81*b
*e^4*(-c^2*x^2+1)^5/c^9/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {200, 5908, 12, 1624, 1813, 1864} \begin {gather*} d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {4 b e^3 \left (1-c^2 x^2\right )^4 \left (9 c^2 d+7 e\right )}{441 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b e^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9 \sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b e \left (1-c^2 x^2\right )^2 \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{945 c^9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right )}{315 c^9 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4*d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) - (4*b*e*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2)^2)/(945*c^9*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*e^2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) - (4*b*e^3*(9*c^2*d + 7*e)*(1 - c^2*x^2)^4)/(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*
e^4*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^4*x*(a + b*ArcCosh[c*x]) + (4*d^3*e*x^3*(a + b*
ArcCosh[c*x]))/3 + (6*d^2*e^2*x^5*(a + b*ArcCosh[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcCosh[c*x]))/7 + (e^4*x^9*(a
+ b*ArcCosh[c*x]))/9

Rule 12

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && 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 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 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 5908

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{315} (b c) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt {-1+c^2 x^2}} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \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 {315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4}{c^8 \sqrt {-1+c^2 x}}+\frac {4 e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \sqrt {-1+c^2 x}}{c^8}+\frac {6 e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (-1+c^2 x\right )^{3/2}}{c^8}+\frac {20 e^3 \left (9 c^2 d+7 e\right ) \left (-1+c^2 x\right )^{5/2}}{c^8}+\frac {35 e^4 \left (-1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt {-1+c x} \sqrt {1+c x}}+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 265, normalized size = 0.67 \begin {gather*} \frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (4480 e^4+320 c^2 e^3 \left (81 d+7 e x^2\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )+8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )\right )}{c^9}+315 b x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right ) \cosh ^{-1}(c x)}{99225} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8) - (b*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(4480*e^4 + 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) + 8*c^6*e*(1
1025*d^3 + 3969*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3*e*x^2 + 23814*d^2*e^2*x
^4 + 8100*d*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6
+ 35*e^4*x^8)*ArcCosh[c*x])/99225

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Maple [A]
time = 1.86, size = 331, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\mathrm {arccosh}\left (c x \right ) d^{4} c^{9} x +\frac {4 \,\mathrm {arccosh}\left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \,\mathrm {arccosh}\left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \,\mathrm {arccosh}\left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\mathrm {arccosh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{4} x^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right )}{99225}\right )}{c^{8}}}{c}\)	\(331\)
default	\(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\mathrm {arccosh}\left (c x \right ) d^{4} c^{9} x +\frac {4 \,\mathrm {arccosh}\left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \,\mathrm {arccosh}\left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \,\mathrm {arccosh}\left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\mathrm {arccosh}\left (c x \right ) e^{4} c^{9} x^{9}}{9}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} e^{4} x^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right )}{99225}\right )}{c^{8}}}{c}\)	\(331\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^8*(d^4*c^9*x+4/3*d^3*c^9*e*x^3+6/5*d^2*c^9*e^2*x^5+4/7*d*c^9*e^3*x^7+1/9*e^4*c^9*x^9)+b/c^8*(arccosh(
c*x)*d^4*c^9*x+4/3*arccosh(c*x)*d^3*c^9*e*x^3+6/5*arccosh(c*x)*d^2*c^9*e^2*x^5+4/7*arccosh(c*x)*d*c^9*e^3*x^7+
1/9*arccosh(c*x)*e^4*c^9*x^9-1/99225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*e^4*x^8+8100*c^8*d*e^3*x^6+23814*c^
8*d^2*e^2*x^4+1400*c^6*e^4*x^6+44100*c^8*d^3*e*x^2+9720*c^6*d*e^3*x^4+99225*c^8*d^4+31752*c^6*d^2*e^2*x^2+1680
*c^4*e^4*x^4+88200*c^6*d^3*e+12960*c^4*d*e^3*x^2+63504*c^4*d^2*e^2+2240*c^2*e^4*x^2+25920*c^2*d*e^3+4480*e^4))
)

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Maxima [A]
time = 0.27, size = 411, normalized size = 1.04 \begin {gather*} \frac {1}{9} \, a x^{9} e^{4} + \frac {4}{7} \, a d x^{7} e^{3} + \frac {6}{5} \, a d^{2} x^{5} e^{2} + \frac {4}{3} \, a d^{3} x^{3} e + a d^{4} x + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} e + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{4}}{c} + \frac {2}{25} \, {\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} e^{2} + \frac {4}{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^{3} + \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^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*x^9*e^4 + 4/7*a*d*x^7*e^3 + 6/5*a*d^2*x^5*e^2 + 4/3*a*d^3*x^3*e + a*d^4*x + 4/9*(3*x^3*arccosh(c*x) - c*
(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d^3*e + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^4/
c + 2/25*(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*e^2 + 4/245*(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^3 + 1/2835*(315*x^9*arccosh(c*x) - (35*sqr
t(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^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1039 vs. \(2 (346) = 692\).
time = 0.36, size = 1039, normalized size = 2.63 \begin {gather*} \frac {11025 \, a c^{9} x^{9} \cosh \left (1\right )^{4} + 11025 \, a c^{9} x^{9} \sinh \left (1\right )^{4} + 56700 \, a c^{9} d x^{7} \cosh \left (1\right )^{3} + 119070 \, a c^{9} d^{2} x^{5} \cosh \left (1\right )^{2} + 132300 \, a c^{9} d^{3} x^{3} \cosh \left (1\right ) + 99225 \, a c^{9} d^{4} x + 6300 \, {\left (7 \, a c^{9} x^{9} \cosh \left (1\right ) + 9 \, a c^{9} d x^{7}\right )} \sinh \left (1\right )^{3} + 1890 \, {\left (35 \, a c^{9} x^{9} \cosh \left (1\right )^{2} + 90 \, a c^{9} d x^{7} \cosh \left (1\right ) + 63 \, a c^{9} d^{2} x^{5}\right )} \sinh \left (1\right )^{2} + 315 \, {\left (35 \, b c^{9} x^{9} \cosh \left (1\right )^{4} + 35 \, b c^{9} x^{9} \sinh \left (1\right )^{4} + 180 \, b c^{9} d x^{7} \cosh \left (1\right )^{3} + 378 \, b c^{9} d^{2} x^{5} \cosh \left (1\right )^{2} + 420 \, b c^{9} d^{3} x^{3} \cosh \left (1\right ) + 315 \, b c^{9} d^{4} x + 20 \, {\left (7 \, b c^{9} x^{9} \cosh \left (1\right ) + 9 \, b c^{9} d x^{7}\right )} \sinh \left (1\right )^{3} + 6 \, {\left (35 \, b c^{9} x^{9} \cosh \left (1\right )^{2} + 90 \, b c^{9} d x^{7} \cosh \left (1\right ) + 63 \, b c^{9} d^{2} x^{5}\right )} \sinh \left (1\right )^{2} + 4 \, {\left (35 \, b c^{9} x^{9} \cosh \left (1\right )^{3} + 135 \, b c^{9} d x^{7} \cosh \left (1\right )^{2} + 189 \, b c^{9} d^{2} x^{5} \cosh \left (1\right ) + 105 \, b c^{9} d^{3} x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 1260 \, {\left (35 \, a c^{9} x^{9} \cosh \left (1\right )^{3} + 135 \, a c^{9} d x^{7} \cosh \left (1\right )^{2} + 189 \, a c^{9} d^{2} x^{5} \cosh \left (1\right ) + 105 \, a c^{9} d^{3} x^{3}\right )} \sinh \left (1\right ) - {\left (99225 \, b c^{8} d^{4} + 35 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )^{4} + 35 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} \sinh \left (1\right )^{4} + 1620 \, {\left (5 \, b c^{8} d x^{6} + 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} + 16 \, b c^{2} d\right )} \cosh \left (1\right )^{3} + 20 \, {\left (405 \, b c^{8} d x^{6} + 486 \, b c^{6} d x^{4} + 648 \, b c^{4} d x^{2} + 1296 \, b c^{2} d + 7 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{3} + 7938 \, {\left (3 \, b c^{8} d^{2} x^{4} + 4 \, b c^{6} d^{2} x^{2} + 8 \, b c^{4} d^{2}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (3969 \, b c^{8} d^{2} x^{4} + 5292 \, b c^{6} d^{2} x^{2} + 10584 \, b c^{4} d^{2} + 35 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )^{2} + 810 \, {\left (5 \, b c^{8} d x^{6} + 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} + 16 \, b c^{2} d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 44100 \, {\left (b c^{8} d^{3} x^{2} + 2 \, b c^{6} d^{3}\right )} \cosh \left (1\right ) + 4 \, {\left (11025 \, b c^{8} d^{3} x^{2} + 22050 \, b c^{6} d^{3} + 35 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} \cosh \left (1\right )^{3} + 1215 \, {\left (5 \, b c^{8} d x^{6} + 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} + 16 \, b c^{2} d\right )} \cosh \left (1\right )^{2} + 3969 \, {\left (3 \, b c^{8} d^{2} x^{4} + 4 \, b c^{6} d^{2} x^{2} + 8 \, b c^{4} d^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(11025*a*c^9*x^9*cosh(1)^4 + 11025*a*c^9*x^9*sinh(1)^4 + 56700*a*c^9*d*x^7*cosh(1)^3 + 119070*a*c^9*d^
2*x^5*cosh(1)^2 + 132300*a*c^9*d^3*x^3*cosh(1) + 99225*a*c^9*d^4*x + 6300*(7*a*c^9*x^9*cosh(1) + 9*a*c^9*d*x^7
)*sinh(1)^3 + 1890*(35*a*c^9*x^9*cosh(1)^2 + 90*a*c^9*d*x^7*cosh(1) + 63*a*c^9*d^2*x^5)*sinh(1)^2 + 315*(35*b*
c^9*x^9*cosh(1)^4 + 35*b*c^9*x^9*sinh(1)^4 + 180*b*c^9*d*x^7*cosh(1)^3 + 378*b*c^9*d^2*x^5*cosh(1)^2 + 420*b*c
^9*d^3*x^3*cosh(1) + 315*b*c^9*d^4*x + 20*(7*b*c^9*x^9*cosh(1) + 9*b*c^9*d*x^7)*sinh(1)^3 + 6*(35*b*c^9*x^9*co
sh(1)^2 + 90*b*c^9*d*x^7*cosh(1) + 63*b*c^9*d^2*x^5)*sinh(1)^2 + 4*(35*b*c^9*x^9*cosh(1)^3 + 135*b*c^9*d*x^7*c
osh(1)^2 + 189*b*c^9*d^2*x^5*cosh(1) + 105*b*c^9*d^3*x^3)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + 1260*(35*a*c
^9*x^9*cosh(1)^3 + 135*a*c^9*d*x^7*cosh(1)^2 + 189*a*c^9*d^2*x^5*cosh(1) + 105*a*c^9*d^3*x^3)*sinh(1) - (99225
*b*c^8*d^4 + 35*(35*b*c^8*x^8 + 40*b*c^6*x^6 + 48*b*c^4*x^4 + 64*b*c^2*x^2 + 128*b)*cosh(1)^4 + 35*(35*b*c^8*x
^8 + 40*b*c^6*x^6 + 48*b*c^4*x^4 + 64*b*c^2*x^2 + 128*b)*sinh(1)^4 + 1620*(5*b*c^8*d*x^6 + 6*b*c^6*d*x^4 + 8*b
*c^4*d*x^2 + 16*b*c^2*d)*cosh(1)^3 + 20*(405*b*c^8*d*x^6 + 486*b*c^6*d*x^4 + 648*b*c^4*d*x^2 + 1296*b*c^2*d +
7*(35*b*c^8*x^8 + 40*b*c^6*x^6 + 48*b*c^4*x^4 + 64*b*c^2*x^2 + 128*b)*cosh(1))*sinh(1)^3 + 7938*(3*b*c^8*d^2*x
^4 + 4*b*c^6*d^2*x^2 + 8*b*c^4*d^2)*cosh(1)^2 + 6*(3969*b*c^8*d^2*x^4 + 5292*b*c^6*d^2*x^2 + 10584*b*c^4*d^2 +
35*(35*b*c^8*x^8 + 40*b*c^6*x^6 + 48*b*c^4*x^4 + 64*b*c^2*x^2 + 128*b)*cosh(1)^2 + 810*(5*b*c^8*d*x^6 + 6*b*c
^6*d*x^4 + 8*b*c^4*d*x^2 + 16*b*c^2*d)*cosh(1))*sinh(1)^2 + 44100*(b*c^8*d^3*x^2 + 2*b*c^6*d^3)*cosh(1) + 4*(1
1025*b*c^8*d^3*x^2 + 22050*b*c^6*d^3 + 35*(35*b*c^8*x^8 + 40*b*c^6*x^6 + 48*b*c^4*x^4 + 64*b*c^2*x^2 + 128*b)*
cosh(1)^3 + 1215*(5*b*c^8*d*x^6 + 6*b*c^6*d*x^4 + 8*b*c^4*d*x^2 + 16*b*c^2*d)*cosh(1)^2 + 3969*(3*b*c^8*d^2*x^
4 + 4*b*c^6*d^2*x^2 + 8*b*c^4*d^2)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/c^9

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Sympy [C] Result contains complex when optimal does not.
time = 1.50, size = 600, normalized size = 1.52 \begin {gather*} \begin {cases} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {6 a d^{2} e^{2} x^{5}}{5} + \frac {4 a d e^{3} x^{7}}{7} + \frac {a e^{4} x^{9}}{9} + b d^{4} x \operatorname {acosh}{\left (c x \right )} + \frac {4 b d^{3} e x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {6 b d^{2} e^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {4 b d e^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} + \frac {b e^{4} x^{9} \operatorname {acosh}{\left (c x \right )}}{9} - \frac {b d^{4} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {4 b d^{3} e x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {6 b d^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {4 b d e^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {b e^{4} x^{8} \sqrt {c^{2} x^{2} - 1}}{81 c} - \frac {8 b d^{3} e \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} - \frac {8 b d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{3}} - \frac {24 b d e^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {8 b e^{4} x^{6} \sqrt {c^{2} x^{2} - 1}}{567 c^{3}} - \frac {16 b d^{2} e^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{5}} - \frac {32 b d e^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {16 b e^{4} x^{4} \sqrt {c^{2} x^{2} - 1}}{945 c^{5}} - \frac {64 b d e^{3} \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} - \frac {64 b e^{4} x^{2} \sqrt {c^{2} x^{2} - 1}}{2835 c^{7}} - \frac {128 b e^{4} \sqrt {c^{2} x^{2} - 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d^{4} x + \frac {4 d^{3} e x^{3}}{3} + \frac {6 d^{2} e^{2} x^{5}}{5} + \frac {4 d e^{3} x^{7}}{7} + \frac {e^{4} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**4*x + 4*a*d**3*e*x**3/3 + 6*a*d**2*e**2*x**5/5 + 4*a*d*e**3*x**7/7 + a*e**4*x**9/9 + b*d**4*x*
acosh(c*x) + 4*b*d**3*e*x**3*acosh(c*x)/3 + 6*b*d**2*e**2*x**5*acosh(c*x)/5 + 4*b*d*e**3*x**7*acosh(c*x)/7 + b
*e**4*x**9*acosh(c*x)/9 - b*d**4*sqrt(c**2*x**2 - 1)/c - 4*b*d**3*e*x**2*sqrt(c**2*x**2 - 1)/(9*c) - 6*b*d**2*
e**2*x**4*sqrt(c**2*x**2 - 1)/(25*c) - 4*b*d*e**3*x**6*sqrt(c**2*x**2 - 1)/(49*c) - b*e**4*x**8*sqrt(c**2*x**2
- 1)/(81*c) - 8*b*d**3*e*sqrt(c**2*x**2 - 1)/(9*c**3) - 8*b*d**2*e**2*x**2*sqrt(c**2*x**2 - 1)/(25*c**3) - 24
*b*d*e**3*x**4*sqrt(c**2*x**2 - 1)/(245*c**3) - 8*b*e**4*x**6*sqrt(c**2*x**2 - 1)/(567*c**3) - 16*b*d**2*e**2*
sqrt(c**2*x**2 - 1)/(25*c**5) - 32*b*d*e**3*x**2*sqrt(c**2*x**2 - 1)/(245*c**5) - 16*b*e**4*x**4*sqrt(c**2*x**
2 - 1)/(945*c**5) - 64*b*d*e**3*sqrt(c**2*x**2 - 1)/(245*c**7) - 64*b*e**4*x**2*sqrt(c**2*x**2 - 1)/(2835*c**7
) - 128*b*e**4*sqrt(c**2*x**2 - 1)/(2835*c**9), Ne(c, 0)), ((a + I*pi*b/2)*(d**4*x + 4*d**3*e*x**3/3 + 6*d**2*
e**2*x**5/5 + 4*d*e**3*x**7/7 + e**4*x**9/9), 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((e*x^2+d)^4*(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 \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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