\(\int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 135 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{75 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d} \]

[Out]

1/5*e^4*(d*x+c)^5*(a+b*arccosh(d*x+c))/d-8/75*b*e^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-4/75*b*e^4*(d*x+c)^2*(d*
x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-1/25*b*e^4*(d*x+c)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5996, 12, 5883, 102, 75} \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d}-\frac {b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{25 d}-\frac {4 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{75 d}-\frac {8 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1}}{75 d} \]

[In]

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

[Out]

(-8*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(75*d) - (4*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c +
d*x])/(75*d) - (b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCo
sh[c + d*x]))/(5*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 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 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 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*} \text {integral}& = \frac {\text {Subst}\left (\int e^4 x^4 (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int x^4 (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d}-\frac {\left (b e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d} \\ & = -\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d}-\frac {\left (b e^4\right ) \text {Subst}\left (\int \frac {4 x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d} \\ & = -\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d}-\frac {\left (4 b e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d} \\ & = -\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d}-\frac {\left (4 b e^4\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{75 d} \\ & = -\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d}-\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{75 d} \\ & = -\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{75 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 (a+b \text {arccosh}(c+d x))}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^4 \left (-\frac {1}{75} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (8+4 (c+d x)^2+3 (c+d x)^4\right )+\frac {1}{5} (c+d x)^5 (a+b \text {arccosh}(c+d x))\right )}{d} \]

[In]

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

[Out]

(e^4*(-1/75*(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(8 + 4*(c + d*x)^2 + 3*(c + d*x)^4)) + ((c + d*x)^5*(a + b
*ArcCosh[c + d*x]))/5))/d

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.58

method	result	size
derivativedivides	\(\frac {\frac {e^{4} a \left (d x +c \right )^{5}}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(78\)
default	\(\frac {\frac {e^{4} a \left (d x +c \right )^{5}}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(78\)
parts	\(\frac {e^{4} a \left (d x +c \right )^{5}}{5 d}+\frac {e^{4} b \left (\frac {\left (d x +c \right )^{5} \operatorname {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d}\)	\(80\)

[In]

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

[Out]

1/d*(1/5*e^4*a*(d*x+c)^5+e^4*b*(1/5*(d*x+c)^5*arccosh(d*x+c)-1/75*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(3*(d*x+c)^4
+4*(d*x+c)^2+8)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (115) = 230\).

Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.07 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {15 \, a d^{5} e^{4} x^{5} + 75 \, a c d^{4} e^{4} x^{4} + 150 \, a c^{2} d^{3} e^{4} x^{3} + 150 \, a c^{3} d^{2} e^{4} x^{2} + 75 \, a c^{4} d e^{4} x + 15 \, {\left (b d^{5} e^{4} x^{5} + 5 \, b c d^{4} e^{4} x^{4} + 10 \, b c^{2} d^{3} e^{4} x^{3} + 10 \, b c^{3} d^{2} e^{4} x^{2} + 5 \, b c^{4} d e^{4} x + b c^{5} e^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (3 \, b d^{4} e^{4} x^{4} + 12 \, b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d e^{4} x + {\left (3 \, b c^{4} + 4 \, b c^{2} + 8 \, b\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{75 \, d} \]

[In]

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

[Out]

1/75*(15*a*d^5*e^4*x^5 + 75*a*c*d^4*e^4*x^4 + 150*a*c^2*d^3*e^4*x^3 + 150*a*c^3*d^2*e^4*x^2 + 75*a*c^4*d*e^4*x
+ 15*(b*d^5*e^4*x^5 + 5*b*c*d^4*e^4*x^4 + 10*b*c^2*d^3*e^4*x^3 + 10*b*c^3*d^2*e^4*x^2 + 5*b*c^4*d*e^4*x + b*c
^5*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (3*b*d^4*e^4*x^4 + 12*b*c*d^3*e^4*x^3 + 2*(9*b*c^2
+ 2*b)*d^2*e^4*x^2 + 4*(3*b*c^3 + 2*b*c)*d*e^4*x + (3*b*c^4 + 4*b*c^2 + 8*b)*e^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2
- 1))/d

Sympy [F]

\[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=e^{4} \left (\int a c^{4}\, dx + \int a d^{4} x^{4}\, dx + \int b c^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 a c d^{3} x^{3}\, dx + \int 6 a c^{2} d^{2} x^{2}\, dx + \int 4 a c^{3} d x\, dx + \int b d^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 b c d^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 6 b c^{2} d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 4 b c^{3} d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

e**4*(Integral(a*c**4, x) + Integral(a*d**4*x**4, x) + Integral(b*c**4*acosh(c + d*x), x) + Integral(4*a*c*d**
3*x**3, x) + Integral(6*a*c**2*d**2*x**2, x) + Integral(4*a*c**3*d*x, x) + Integral(b*d**4*x**4*acosh(c + d*x)
, x) + Integral(4*b*c*d**3*x**3*acosh(c + d*x), x) + Integral(6*b*c**2*d**2*x**2*acosh(c + d*x), x) + Integral
(4*b*c**3*d*x*acosh(c + d*x), x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (115) = 230\).

Time = 0.23 (sec) , antiderivative size = 1241, normalized size of antiderivative = 9.19 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/5*a*d^4*e^4*x^5 + a*c*d^3*e^4*x^4 + 2*a*c^2*d^2*e^4*x^3 + 2*a*c^3*d*e^4*x^2 + (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))*b*c^3*d*e^4 + 1/3*(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*log(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))*b*c^2*d^2*e^4 + 1/24*(24*x^4*arc
cosh(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
+ 105*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)*b*c*d^3*e^4 + 1/600*(120*x^5*arccosh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^4/d^2 - 54*s
qrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^3/d^3 + 126*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^2*x^2/d^4 - 945*c^5*log(2
*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^6 - 315*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^3*x/d^5
- 32*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x^2/d^4 + 1050*(c^2 - 1)*c^3*log(2*d^2*x + 2*c*d + 2*sqrt(d^2
*x^2 + 2*c*d*x + c^2 - 1)*d)/d^6 + 945*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^4/d^6 + 161*sqrt(d^2*x^2 + 2*c*d*x
+ c^2 - 1)*(c^2 - 1)*c*x/d^5 - 225*(c^2 - 1)^2*c*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/
d^6 - 735*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c^2/d^6 + 64*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)
^2/d^6)*d)*b*d^4*e^4 + a*c^4*e^4*x + ((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*b*c^4*e^4/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (115) = 230\).

Time = 0.99 (sec) , antiderivative size = 846, normalized size of antiderivative = 6.27 \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{5} \, a d^{4} e^{4} x^{5} + a c d^{3} e^{4} x^{4} + 2 \, a c^{2} d^{2} e^{4} x^{3} + 2 \, a c^{3} d e^{4} x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c^{4} e^{4} + {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c^{3} d e^{4} + \frac {1}{3} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d + 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} + 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b c^{2} d^{2} e^{4} + \frac {1}{24} \, {\left (24 \, x^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{d^{2}} - \frac {7 \, c}{d^{3}}\right )} + \frac {26 \, c^{2} d^{3} + 9 \, d^{3}}{d^{7}}\right )} x - \frac {5 \, {\left (10 \, c^{3} d^{2} + 11 \, c d^{2}\right )}}{d^{7}}\right )} - \frac {3 \, {\left (8 \, c^{4} + 24 \, c^{2} + 3\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{4} {\left | d \right |}}\right )} d\right )} b c d^{3} e^{4} + \frac {1}{600} \, {\left (120 \, x^{5} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (2 \, {\left (3 \, x {\left (\frac {4 \, x}{d^{2}} - \frac {9 \, c}{d^{3}}\right )} + \frac {47 \, c^{2} d^{5} + 16 \, d^{5}}{d^{9}}\right )} x - \frac {7 \, {\left (22 \, c^{3} d^{4} + 23 \, c d^{4}\right )}}{d^{9}}\right )} x + \frac {274 \, c^{4} d^{3} + 607 \, c^{2} d^{3} + 64 \, d^{3}}{d^{9}}\right )} + \frac {15 \, {\left (8 \, c^{5} + 40 \, c^{3} + 15 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{5} {\left | d \right |}}\right )} d\right )} b d^{4} e^{4} + a c^{4} e^{4} x \]

[In]

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

[Out]

1/5*a*d^4*e^4*x^5 + a*c*d^3*e^4*x^4 + 2*a*c^2*d^2*e^4*x^3 + 2*a*c^3*d*e^4*x^2 - (d*(c*log(abs(-c*d - (x*abs(d)
- sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d*abs(d)) + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)/d^2) - x*log(d*x
+ c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)))*b*c^4*e^4 + (2*x^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))
- (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x/d^2 - 3*c/d^3) - (2*c^2 + 1)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2
+ 2*c*d*x + c^2 - 1))*abs(d)))/(d^2*abs(d)))*d)*b*c^3*d*e^4 + 1/3*(6*x^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x
+ c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x*(2*x/d^2 - 5*c/d^3) + (11*c^2*d + 4*d)/d^5) + 3*(2*c^3 + 3
*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^3*abs(d)))*d)*b*c^2*d^2*e^4 + 1/
24*(24*x^4*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*((2*x*(3*x/d^
2 - 7*c/d^3) + (26*c^2*d^3 + 9*d^3)/d^7)*x - 5*(10*c^3*d^2 + 11*c*d^2)/d^7) - 3*(8*c^4 + 24*c^2 + 3)*log(abs(-
c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^4*abs(d)))*d)*b*c*d^3*e^4 + 1/600*(120*x^5*lo
g(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*((2*(3*x*(4*x/d^2 - 9*c/d^
3) + (47*c^2*d^5 + 16*d^5)/d^9)*x - 7*(22*c^3*d^4 + 23*c*d^4)/d^9)*x + (274*c^4*d^3 + 607*c^2*d^3 + 64*d^3)/d^
9) + 15*(8*c^5 + 40*c^3 + 15*c)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^5*ab
s(d)))*d)*b*d^4*e^4 + a*c^4*e^4*x

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^4 (a+b \text {arccosh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^4\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]

[In]

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

[Out]

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