\(\int (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [483]

Optimal result

Integrand size = 18, antiderivative size = 287 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \left (1-c^2 x^2\right )}{35 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^2}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x)) \]

[Out]

d^3*x*(a+b*arccosh(c*x))+d^2*e*x^3*(a+b*arccosh(c*x))+3/5*d*e^2*x^5*(a+b*arccosh(c*x))+1/7*e^3*x^7*(a+b*arccos
h(c*x))+1/35*b*(35*c^6*d^3+35*c^4*d^2*e+21*c^2*d*e^2+5*e^3)*(-c^2*x^2+1)/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/105
*b*e*(35*c^4*d^2+42*c^2*d*e+15*e^2)*(-c^2*x^2+1)^2/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/175*b*e^2*(7*c^2*d+5*e)*(
-c^2*x^2+1)^3/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/49*b*e^3*(-c^2*x^2+1)^4/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {200, 5908, 12, 1624, 1813, 1864} \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))+\frac {3 b e^2 \left (1-c^2 x^2\right )^3 \left (7 c^2 d+5 e\right )}{175 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right )}{35 c^7 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*(1 - c^2*x^2))/(35*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
b*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2)^2)/(105*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*e^2*(7*c
^2*d + 5*e)*(1 - c^2*x^2)^3)/(175*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(1 - c^2*x^2)^4)/(49*c^7*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) + d^2*e*x^3*(a + b*ArcCosh[c*x]) + (3*d*e^2*x^5*(a + b*ArcC
osh[c*x]))/5 + (e^3*x^7*(a + b*ArcCosh[c*x]))/7

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*} \text {integral}& = d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))-(b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))-\frac {1}{35} (b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3}{c^6 \sqrt {-1+c^2 x}}+\frac {e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {3 e^2 \left (7 c^2 d+5 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {5 e^3 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \left (1-c^2 x^2\right )}{35 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^2}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+d^3 x (a+b \text {arccosh}(c x))+d^2 e x^3 (a+b \text {arccosh}(c x))+\frac {3}{5} d e^2 x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^3 x^7 (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.67 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=a \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )}{3675 c^7}+\frac {1}{35} b x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \text {arccosh}(c x) \]

[In]

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

[Out]

a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*
(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*
x^4 + 75*e^3*x^6)))/(3675*c^7) + (b*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcCosh[c*x])/35

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.74

method	result	size
parts	\(a \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \,\operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{5}}{5}+c \,\operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{3}+\operatorname {arccosh}\left (c x \right ) c x \,d^{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675 c^{6}}\right )}{c}\)	\(211\)
derivativedivides	\(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\)	\(235\)
default	\(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d^{3} c^{7} x +\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} x^{4} e^{3}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} x^{2} e^{3}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\)	\(235\)

[In]

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

[Out]

a*(1/7*e^3*x^7+3/5*d*e^2*x^5+d^2*e*x^3+d^3*x)+b/c*(1/7*c*arccosh(c*x)*e^3*x^7+3/5*c*arccosh(c*x)*d*e^2*x^5+c*a
rccosh(c*x)*d^2*e*x^3+arccosh(c*x)*c*x*d^3-1/3675/c^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*e^3*x^6+441*c^6*d*e^
2*x^4+1225*c^6*d^2*e*x^2+90*c^4*e^3*x^4+3675*c^6*d^3+588*c^4*d*e^2*x^2+2450*c^4*d^2*e+120*c^2*e^3*x^2+1176*c^2
*d*e^2+240*e^3))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.84 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \, {\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} + 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \, {\left (49 \, b c^{6} d e^{2} + 10 \, b c^{4} e^{3}\right )} x^{4} + 240 \, b e^{3} + {\left (1225 \, b c^{6} d^{2} e + 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{7}} \]

[In]

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

[Out]

1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*
x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^2*e*x^3 + 35*b*c^7*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*e^3*x
^6 + 3675*b*c^6*d^3 + 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^2 + 10*b*c^4*e^3)*x^4 + 240*b*e^3
+ (1225*b*c^6*d^2*e + 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))/c^7

Sympy [F]

\[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a e^{3} x^{7} + \frac {3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((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

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]

[In]

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

[Out]

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