3.2.14 \(\int (c e+d e x)^3 (a+b \cosh ^{-1}(c+d x))^3 \, dx\) [114]

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

[Out]

-45/256*b^3*e^3*arccosh(d*x+c)/d+9/32*b^2*e^3*(d*x+c)^2*(a+b*arccosh(d*x+c))/d+3/32*b^2*e^3*(d*x+c)^4*(a+b*arc
cosh(d*x+c))/d-3/32*e^3*(a+b*arccosh(d*x+c))^3/d+1/4*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^3/d-45/256*b^3*e^3*(d*
x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-3/128*b^3*e^3*(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-9/32*b*e^3*(d
*x+c)*(a+b*arccosh(d*x+c))^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-3/16*b*e^3*(d*x+c)^3*(a+b*arccosh(d*x+c))^2*(d*
x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]
time = 0.41, antiderivative size = 307, normalized size of antiderivative = 1.00, 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 = {5996, 12, 5883, 5939, 5893, 92, 54, 102} \begin {gather*} \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} \end {gather*}

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 12

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

Rule 54

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 92

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 102

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]

Rule 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5893

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

Rule 5939

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

Rule 5996

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
]

Rubi steps

\begin {align*} \int (c e+d e x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.33, size = 359, normalized size = 1.17 \begin {gather*} \frac {e^3 \left (72 a b^2 (c+d x)^2+8 a \left (8 a^2+3 b^2\right ) (c+d x)^4+3 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (-3 \left (8 a^2+5 b^2\right )-2 \left (8 a^2+b^2\right ) (c+d x)^2\right )-24 b (c+d x) \left (-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}\right ) \cosh ^{-1}(c+d x)+24 b^2 \left (-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}\right ) \cosh ^{-1}(c+d x)^2+8 b^3 \left (-3+8 (c+d x)^4\right ) \cosh ^{-1}(c+d x)^3-9 b \left (8 a^2+5 b^2\right ) \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{256 d} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(646\) vs. \(2(273)=546\).
time = 41.01, size = 647, normalized size = 2.11

method	result	size
default	\(\frac {e^{3} \left (d x +c \right )^{4} a^{3}}{4 d}-\frac {b^{3} e^{3} \left (-32 \left (\cosh ^{2}\left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )\right ) \mathrm {arccosh}\left (d x +c \right )^{3}+24 \sinh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right ) \cosh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right ) \mathrm {arccosh}\left (d x +c \right )^{2}-64 \cosh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right ) \mathrm {arccosh}\left (d x +c \right )^{3}+96 \mathrm {arccosh}\left (d x +c \right )^{2} \sinh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )-12 \,\mathrm {arccosh}\left (d x +c \right ) \left (\cosh ^{2}\left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )\right )+16 \mathrm {arccosh}\left (d x +c \right )^{3}+3 \sinh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right ) \cosh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )-96 \cosh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right ) \mathrm {arccosh}\left (d x +c \right )+48 \sinh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )+6 \,\mathrm {arccosh}\left (d x +c \right )\right )}{512 d}+\frac {3 a \,b^{2} e^{3} \left (\frac {\left (8 \mathrm {arccosh}\left (d x +c \right )^{2}+1\right ) \left (\cosh ^{2}\left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )\right )}{128}+\frac {\left (4 \mathrm {arccosh}\left (d x +c \right )^{2}-\mathrm {arccosh}\left (d x +c \right ) \sinh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )+2\right ) \cosh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )}{32}-\frac {\mathrm {arccosh}\left (d x +c \right )^{2}}{32}-\frac {\mathrm {arccosh}\left (d x +c \right ) \sinh \left (2 \,\mathrm {arccosh}\left (d x +c \right )\right )}{8}-\frac {1}{256}\right )}{d}+\frac {3 a^{2} b \,e^{3} d^{3} \mathrm {arccosh}\left (d x +c \right ) x^{4}}{4}+3 a^{2} b \,e^{3} d^{2} \mathrm {arccosh}\left (d x +c \right ) x^{3} c +\frac {9 a^{2} b \,e^{3} d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} c^{2}}{2}+3 a^{2} b \,e^{3} \mathrm {arccosh}\left (d x +c \right ) x \,c^{3}+\frac {3 a^{2} b \,e^{3} \mathrm {arccosh}\left (d x +c \right ) c^{4}}{4 d}-\frac {3 a^{2} b \,e^{3} d^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{3}}{16}-\frac {9 a^{2} b \,e^{3} d \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x^{2} c}{16}-\frac {9 a^{2} b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x \,c^{2}}{16}-\frac {3 a^{2} b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c^{3}}{16 d}-\frac {9 a^{2} b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x}{32}-\frac {9 a^{2} b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, c}{32 d}-\frac {9 a^{2} b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{32 d \sqrt {\left (d x +c \right )^{2}-1}}\)	\(647\)

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,method=_RETURNVERBOSE)

[Out]

1/4*e^3*(d*x+c)^4*a^3/d-1/512*b^3*e^3*(-32*cosh(2*arccosh(d*x+c))^2*arccosh(d*x+c)^3+24*sinh(2*arccosh(d*x+c))
*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)^2-64*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)^3+96*arccosh(d*x+c)^2*sinh(2
*arccosh(d*x+c))-12*arccosh(d*x+c)*cosh(2*arccosh(d*x+c))^2+16*arccosh(d*x+c)^3+3*sinh(2*arccosh(d*x+c))*cosh(
2*arccosh(d*x+c))-96*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)+48*sinh(2*arccosh(d*x+c))+6*arccosh(d*x+c))/d+3*a*b
^2*e^3/d*(1/128*(8*arccosh(d*x+c)^2+1)*cosh(2*arccosh(d*x+c))^2+1/32*(4*arccosh(d*x+c)^2-arccosh(d*x+c)*sinh(2
*arccosh(d*x+c))+2)*cosh(2*arccosh(d*x+c))-1/32*arccosh(d*x+c)^2-1/8*arccosh(d*x+c)*sinh(2*arccosh(d*x+c))-1/2
56)+3/4*a^2*b*e^3*d^3*arccosh(d*x+c)*x^4+3*a^2*b*e^3*d^2*arccosh(d*x+c)*x^3*c+9/2*a^2*b*e^3*d*arccosh(d*x+c)*x
^2*c^2+3*a^2*b*e^3*arccosh(d*x+c)*x*c^3+3/4*a^2*b*e^3/d*arccosh(d*x+c)*c^4-3/16*a^2*b*e^3*d^2*(d*x+c-1)^(1/2)*
(d*x+c+1)^(1/2)*x^3-9/16*a^2*b*e^3*d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-9/16*a^2*b*e^3*(d*x+c-1)^(1/2)*(d*x
+c+1)^(1/2)*x*c^2-3/16*a^2*b*e^3/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-9/32*a^2*b*e^3*(d*x+c-1)^(1/2)*(d*x+c+1
)^(1/2)*x-9/32*a^2*b*e^3/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-9/32*a^2*b*e^3/d*(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))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/4*a^3*d^3*x^4*e^3 + a^3*c*d^2*x^3*e^3 + 3/2*a^3*c^2*d*x^2*e^3 + 9/4*(2*x^2*arccosh(d*x + c) - d*(3*c^2*log(2
*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 -
1)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d
^3))*a^2*b*c^2*d*e^3 + 1/2*(6*x^3*arccosh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*l
og(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3
+ 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x +
c^2 - 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a^2*b*c*d^2*e^3 + 1/32*(24*x^4*arccosh
(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^2/d^3 + 10
5*c^4*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*
c^2*x/d^4 - 90*(c^2 - 1)*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 - 105*sqrt(d^2*x
^2 + 2*c*d*x + c^2 - 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*log(2*d^
2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c/d^
5)*d)*a^2*b*d^3*e^3 + a^3*c^3*x*e^3 + 3*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^2*b*c^3*e^3/d +
1/4*(b^3*d^3*x^4*e^3 + 4*b^3*c*d^2*x^3*e^3 + 6*b^3*c^2*d*x^2*e^3 + 4*b^3*c^3*x*e^3)*log(d*x + sqrt(d*x + c +
1)*sqrt(d*x + c - 1) + c)^3 + integrate(3/4*((4*a*b^2*d^6 - b^3*d^6)*x^6*e^3 + 6*(4*a*b^2*c*d^5 - b^3*c*d^5)*x
^5*e^3 + (4*(15*c^2*d^4 - d^4)*a*b^2 - (15*c^2*d^4 - d^4)*b^3)*x^4*e^3 + 4*(c^6 - c^4)*a*b^2*e^3 + 4*(4*(5*c^3
*d^3 - c*d^3)*a*b^2 - (5*c^3*d^3 - c*d^3)*b^3)*x^3*e^3 + 2*(6*(5*c^4*d^2 - 2*c^2*d^2)*a*b^2 - (7*c^4*d^2 - 3*c
^2*d^2)*b^3)*x^2*e^3 + 4*(2*(3*c^5*d - 2*c^3*d)*a*b^2 - (c^5*d - c^3*d)*b^3)*x*e^3 + ((4*a*b^2*d^5 - b^3*d^5)*
x^5*e^3 + 5*(4*a*b^2*c*d^4 - b^3*c*d^4)*x^4*e^3 + 4*(c^5 - c^3)*a*b^2*e^3 - 2*(5*b^3*c^2*d^3 - 2*(10*c^2*d^3 -
d^3)*a*b^2)*x^3*e^3 - 2*(5*b^3*c^3*d^2 - 2*(10*c^3*d^2 - 3*c*d^2)*a*b^2)*x^2*e^3 - 4*(b^3*c^4*d - (5*c^4*d -
3*c^2*d)*a*b^2)*x*e^3)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)
^2/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*c^2*d
- d)*x - c), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2620 vs. \(2 (264) = 528\).
time = 0.43, size = 2620, normalized size = 8.53 \begin {gather*} \text {Too large to display} \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1828 vs. \(2 (294) = 588\).
time = 0.99, size = 1828, normalized size = 5.95 \begin {gather*} \text {Too large to display} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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