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

Optimal. Leaf size=307 \[ \frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{3 b e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{9 b e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{32 d}-\frac{3 b^3 e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3}{128 d}-\frac{45 b^3 e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{256 d}-\frac{45 b^3 e^3 \cosh ^{-1}(c+d x)}{256 d} \]

[Out]

(-45*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(256*d) - (3*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)
^3*Sqrt[1 + c + d*x])/(128*d) - (45*b^3*e^3*ArcCosh[c + d*x])/(256*d) + (9*b^2*e^3*(c + d*x)^2*(a + b*ArcCosh[
c + d*x]))/(32*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x]))/(32*d) - (9*b*e^3*Sqrt[-1 + c + d*x]*(c +
d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(32*d) - (3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 +
c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a
+ b*ArcCosh[c + d*x])^3)/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.6246, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5866, 12, 5662, 5759, 5676, 90, 52, 100} \[ \frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{3 b e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{9 b e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{32 d}-\frac{3 b^3 e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3}{128 d}-\frac{45 b^3 e^3 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{256 d}-\frac{45 b^3 e^3 \cosh ^{-1}(c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(256*d) - (3*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)
^3*Sqrt[1 + c + d*x])/(128*d) - (45*b^3*e^3*ArcCosh[c + d*x])/(256*d) + (9*b^2*e^3*(c + d*x)^2*(a + b*ArcCosh[
c + d*x]))/(32*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x]))/(32*d) - (9*b*e^3*Sqrt[-1 + c + d*x]*(c +
d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(32*d) - (3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 +
c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a
+ b*ArcCosh[c + d*x])^3)/(4*d)

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 12

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

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rubi steps

\begin{align*} \int (c e+d e x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \cosh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (3 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac{3 b e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (9 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{16 d}+\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{8 d}\\ &=\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{9 b e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 b e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (9 b e^3\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{32 d}+\frac{\left (9 b^2 e^3\right ) \operatorname{Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{16 d}-\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac{3 b^3 e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{128 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{9 b e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 b e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{3 x^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{128 d}-\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac{9 b^3 e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{64 d}-\frac{3 b^3 e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{128 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{9 b e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 b e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{128 d}-\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac{45 b^3 e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{256 d}-\frac{3 b^3 e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{128 d}-\frac{9 b^3 e^3 \cosh ^{-1}(c+d x)}{64 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{9 b e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 b e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{\left (9 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{256 d}\\ &=-\frac{45 b^3 e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{256 d}-\frac{3 b^3 e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{128 d}-\frac{45 b^3 e^3 \cosh ^{-1}(c+d x)}{256 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}+\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{9 b e^3 \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{32 d}-\frac{3 b e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.525829, size = 359, normalized size = 1.17 \[ \frac{e^3 \left (8 a \left (8 a^2+3 b^2\right ) (c+d x)^4+3 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \left (-2 \left (8 a^2+b^2\right ) (c+d x)^2-3 \left (8 a^2+5 b^2\right )\right )-9 b \left (8 a^2+5 b^2\right ) \log \left (\sqrt{c+d x-1} \sqrt{c+d x+1}+c+d x\right )-24 b (c+d x) \cosh ^{-1}(c+d x) \left (-8 a^2 (c+d x)^3+4 a b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2+6 a b \sqrt{c+d x-1} \sqrt{c+d x+1}-b^2 (c+d x)^3-3 b^2 (c+d x)\right )+72 a b^2 (c+d x)^2+24 b^2 \cosh ^{-1}(c+d x)^2 \left (8 a (c+d x)^4-3 a-2 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3-3 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)\right )+8 b^3 \left (8 (c+d x)^4-3\right ) \cosh ^{-1}(c+d x)^3\right )}{256 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^3*(72*a*b^2*(c + d*x)^2 + 8*a*(8*a^2 + 3*b^2)*(c + d*x)^4 + 3*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d
*x]*(-3*(8*a^2 + 5*b^2) - 2*(8*a^2 + b^2)*(c + d*x)^2) - 24*b*(c + d*x)*(-3*b^2*(c + d*x) - 8*a^2*(c + d*x)^3
- b^2*(c + d*x)^3 + 6*a*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + 4*a*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 +
c + d*x])*ArcCosh[c + d*x] + 24*b^2*(-3*a + 8*a*(c + d*x)^4 - 3*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d
*x] - 2*b*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 + 8*b^3*(-3 + 8*(c + d*x)^4)*Ar
cCosh[c + d*x]^3 - 9*b*(8*a^2 + 5*b^2)*Log[c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]]))/(256*d)

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Maple [B]  time = 0.044, size = 1554, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/16*d*e^3*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-3/8*d^2*e^3*a*b^2*arccosh(d*x+c)*(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2)*x^3-9/32/d*e^3*a^2*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/((d*x+c)^2-1)^(1/2)*ln(d*x+c+((
d*x+c)^2-1)^(1/2))-3/8/d*e^3*a*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-9/16/d*e^3*a*b^2*arccosh
(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-9/16*d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*a^2*b*c*e^3-9/8*e^3*a*b^2
*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-9/8*d*e^3*a*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)
^(1/2)*x^2*c-45/256*b^3*e^3*arccosh(d*x+c)/d+9/32*d*e^3*b^3*arccosh(d*x+c)*x^2+3/8*e^3*b^3*arccosh(d*x+c)*x*c^
3+9/16*e^3*b^3*arccosh(d*x+c)*x*c-45/256*e^3*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x+9/32*d*e^3*a*b^2*x^2+3/32*d
^3*e^3*a*b^2*x^4+3/2*d*x^2*a^3*c^2*e^3+d^2*x^3*a^3*c*e^3+e^3*b^3*arccosh(d*x+c)^3*x*c^3+1/4/d*e^3*b^3*arccosh(
d*x+c)^3*c^4+3/32/d*e^3*b^3*arccosh(d*x+c)*c^4+9/32/d*e^3*b^3*arccosh(d*x+c)*c^2+9/16*e^3*a*b^2*x*c+3/8*e^3*a*
b^2*x*c^3+1/4*d^3*e^3*b^3*arccosh(d*x+c)^3*x^4+3/32*d^3*e^3*b^3*arccosh(d*x+c)*x^4+9/32/d*e^3*a*b^2*c^2+3/32/d
*e^3*a*b^2*c^4+d^2*e^3*b^3*arccosh(d*x+c)^3*x^3*c+3/4/d*e^3*a*b^2*arccosh(d*x+c)^2*c^4-3/128/d*e^3*b^3*(d*x+c-
1)^(1/2)*(d*x+c+1)^(1/2)*c^3-45/256/d*e^3*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-3/128*d^2*e^3*b^3*(d*x+c-1)^(1
/2)*(d*x+c+1)^(1/2)*x^3+3/4*d^3*e^3*a*b^2*arccosh(d*x+c)^2*x^4+3/8*d^2*e^3*b^3*arccosh(d*x+c)*x^3*c+9/16*d*e^3
*a*b^2*x^2*c^2-9/32*e^3*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x-9/32*(d*x+c-1)^(1/2)*(d*x+c+1)^
(1/2)*x*a^2*b*e^3+3*arccosh(d*x+c)*x*a^2*b*c^3*e^3-9/128*e^3*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2+3/4/d*a
rccosh(d*x+c)*a^2*b*c^4*e^3+9/16*d*e^3*b^3*arccosh(d*x+c)*x^2*c^2+3/8*d^2*e^3*a*b^2*x^3*c+3/2*d*e^3*b^3*arccos
h(d*x+c)^3*x^2*c^2+1/4/d*a^3*c^4*e^3+x*a^3*c^3*e^3+1/4*d^3*x^4*a^3*e^3-3/32/d*e^3*b^3*arccosh(d*x+c)^3-9/32/d*
e^3*a*b^2*arccosh(d*x+c)^2+3/4*d^3*arccosh(d*x+c)*x^4*a^2*b*e^3+3*e^3*a*b^2*arccosh(d*x+c)^2*x*c^3-9/16*e^3*b^
3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-3/16*d^2*e^3*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*
x+c+1)^(1/2)*x^3-3/16*d^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^3*a^2*b*e^3-3/16/d*e^3*b^3*arccosh(d*x+c)^2*(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-9/32/d*e^3*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-3/16/d*(d*x+c-
1)^(1/2)*(d*x+c+1)^(1/2)*a^2*b*c^3*e^3-9/32/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*a^2*b*c*e^3+3*d^2*arccosh(d*x+c)
*x^3*a^2*b*c*e^3+9/2*d*arccosh(d*x+c)*x^2*a^2*b*c^2*e^3+3*d^2*e^3*a*b^2*arccosh(d*x+c)^2*x^3*c+9/2*d*e^3*a*b^2
*arccosh(d*x+c)^2*x^2*c^2-9/16*e^3*a*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x-9/128*d*e^3*b^3*(d*x
+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-9/16*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*a^2*b*c^2*e^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.66008, size = 1773, normalized size = 5.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(8*(8*a^3 + 3*a*b^2)*d^4*e^3*x^4 + 32*(8*a^3 + 3*a*b^2)*c*d^3*e^3*x^3 + 24*(3*a*b^2 + 2*(8*a^3 + 3*a*b^2
)*c^2)*d^2*e^3*x^2 + 16*(9*a*b^2*c + 2*(8*a^3 + 3*a*b^2)*c^3)*d*e^3*x + 8*(8*b^3*d^4*e^3*x^4 + 32*b^3*c*d^3*e^
3*x^3 + 48*b^3*c^2*d^2*e^3*x^2 + 32*b^3*c^3*d*e^3*x + (8*b^3*c^4 - 3*b^3)*e^3)*log(d*x + c + sqrt(d^2*x^2 + 2*
c*d*x + c^2 - 1))^3 + 24*(8*a*b^2*d^4*e^3*x^4 + 32*a*b^2*c*d^3*e^3*x^3 + 48*a*b^2*c^2*d^2*e^3*x^2 + 32*a*b^2*c
^3*d*e^3*x + (8*a*b^2*c^4 - 3*a*b^2)*e^3 - (2*b^3*d^3*e^3*x^3 + 6*b^3*c*d^2*e^3*x^2 + 3*(2*b^3*c^2 + b^3)*d*e^
3*x + (2*b^3*c^3 + 3*b^3*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2
- 1))^2 + 3*(8*(8*a^2*b + b^3)*d^4*e^3*x^4 + 32*(8*a^2*b + b^3)*c*d^3*e^3*x^3 + 24*(b^3 + 2*(8*a^2*b + b^3)*c
^2)*d^2*e^3*x^2 + 16*(3*b^3*c + 2*(8*a^2*b + b^3)*c^3)*d*e^3*x + (24*b^3*c^2 + 8*(8*a^2*b + b^3)*c^4 - 24*a^2*
b - 15*b^3)*e^3 - 16*(2*a*b^2*d^3*e^3*x^3 + 6*a*b^2*c*d^2*e^3*x^2 + 3*(2*a*b^2*c^2 + a*b^2)*d*e^3*x + (2*a*b^2
*c^3 + 3*a*b^2*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 3
*(2*(8*a^2*b + b^3)*d^3*e^3*x^3 + 6*(8*a^2*b + b^3)*c*d^2*e^3*x^2 + 3*(8*a^2*b + 5*b^3 + 2*(8*a^2*b + b^3)*c^2
)*d*e^3*x + (2*(8*a^2*b + b^3)*c^3 + 3*(8*a^2*b + 5*b^3)*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

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Sympy [A]  time = 13.486, size = 1828, normalized size = 5.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*c**3*e**3*x + 3*a**3*c**2*d*e**3*x**2/2 + a**3*c*d**2*e**3*x**3 + a**3*d**3*e**3*x**4/4 + 3*a*
*2*b*c**4*e**3*acosh(c + d*x)/(4*d) + 3*a**2*b*c**3*e**3*x*acosh(c + d*x) - 3*a**2*b*c**3*e**3*sqrt(c**2 + 2*c
*d*x + d**2*x**2 - 1)/(16*d) + 9*a**2*b*c**2*d*e**3*x**2*acosh(c + d*x)/2 - 9*a**2*b*c**2*e**3*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 - 1)/16 + 3*a**2*b*c*d**2*e**3*x**3*acosh(c + d*x) - 9*a**2*b*c*d*e**3*x**2*sqrt(c**2 + 2*c
*d*x + d**2*x**2 - 1)/16 - 9*a**2*b*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(32*d) + 3*a**2*b*d**3*e**3*x*
*4*acosh(c + d*x)/4 - 3*a**2*b*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/16 - 9*a**2*b*e**3*x*sqrt(c
**2 + 2*c*d*x + d**2*x**2 - 1)/32 - 9*a**2*b*e**3*acosh(c + d*x)/(32*d) + 3*a*b**2*c**4*e**3*acosh(c + d*x)**2
/(4*d) + 3*a*b**2*c**3*e**3*x*acosh(c + d*x)**2 + 3*a*b**2*c**3*e**3*x/8 - 3*a*b**2*c**3*e**3*sqrt(c**2 + 2*c*
d*x + d**2*x**2 - 1)*acosh(c + d*x)/(8*d) + 9*a*b**2*c**2*d*e**3*x**2*acosh(c + d*x)**2/2 + 9*a*b**2*c**2*d*e*
*3*x**2/16 - 9*a*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/8 + 3*a*b**2*c*d**2*e**3
*x**3*acosh(c + d*x)**2 + 3*a*b**2*c*d**2*e**3*x**3/8 - 9*a*b**2*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2
- 1)*acosh(c + d*x)/8 + 9*a*b**2*c*e**3*x/16 - 9*a*b**2*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c +
d*x)/(16*d) + 3*a*b**2*d**3*e**3*x**4*acosh(c + d*x)**2/4 + 3*a*b**2*d**3*e**3*x**4/32 - 3*a*b**2*d**2*e**3*x
**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/8 + 9*a*b**2*d*e**3*x**2/32 - 9*a*b**2*e**3*x*sqrt(c**
2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/16 - 9*a*b**2*e**3*acosh(c + d*x)**2/(32*d) + b**3*c**4*e**3*acosh
(c + d*x)**3/(4*d) + 3*b**3*c**4*e**3*acosh(c + d*x)/(32*d) + b**3*c**3*e**3*x*acosh(c + d*x)**3 + 3*b**3*c**3
*e**3*x*acosh(c + d*x)/8 - 3*b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/(16*d) - 3*
b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(128*d) + 3*b**3*c**2*d*e**3*x**2*acosh(c + d*x)**3/2 + 9*
b**3*c**2*d*e**3*x**2*acosh(c + d*x)/16 - 9*b**3*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*
x)**2/16 - 9*b**3*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/128 + 9*b**3*c**2*e**3*acosh(c + d*x)/(32*d
) + b**3*c*d**2*e**3*x**3*acosh(c + d*x)**3 + 3*b**3*c*d**2*e**3*x**3*acosh(c + d*x)/8 - 9*b**3*c*d*e**3*x**2*
sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/16 - 9*b**3*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x*
*2 - 1)/128 + 9*b**3*c*e**3*x*acosh(c + d*x)/16 - 9*b**3*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c +
d*x)**2/(32*d) - 45*b**3*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(256*d) + b**3*d**3*e**3*x**4*acosh(c +
d*x)**3/4 + 3*b**3*d**3*e**3*x**4*acosh(c + d*x)/32 - 3*b**3*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 -
1)*acosh(c + d*x)**2/16 - 3*b**3*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/128 + 9*b**3*d*e**3*x**2*
acosh(c + d*x)/32 - 9*b**3*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/32 - 45*b**3*e**3*x*s
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/256 - 3*b**3*e**3*acosh(c + d*x)**3/(32*d) - 45*b**3*e**3*acosh(c + d*x)/(
256*d), Ne(d, 0)), (c**3*e**3*x*(a + b*acosh(c))**3, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{3}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^3, x)