\(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx\) [124]

Optimal result

Integrand size = 21, antiderivative size = 209 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{2 d}-\frac {3 b^2 e (a+b \text {arccosh}(c+d x))^2}{4 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}-\frac {e (a+b \text {arccosh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883, 5939, 5893, 30} \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=-\frac {3 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {3 b^2 e (a+b \text {arccosh}(c+d x))^2}{4 d}-\frac {b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d}-\frac {e (a+b \text {arccosh}(c+d x))^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \]

[In]

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

[Out]

(3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])
)/(2*d) - (3*b^2*e*(a + b*ArcCosh[c + d*x])^2)/(4*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/(2*d)
- (b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/d - (e*(a + b*ArcCosh[c + d*
x])^4)/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^4)/(2*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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*} \text {integral}& = \frac {\text {Subst}\left (\int e x (a+b \text {arccosh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \text {arccosh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d}-\frac {(2 b e) \text {Subst}\left (\int \frac {x^2 (a+b \text {arccosh}(x))^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int x (a+b \text {arccosh}(x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}-\frac {e (a+b \text {arccosh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d}-\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 (a+b \text {arccosh}(x))}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{2 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}-\frac {e (a+b \text {arccosh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d}-\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d} \\ & = \frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{2 d}-\frac {3 b^2 e (a+b \text {arccosh}(c+d x))^2}{4 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d}-\frac {e (a+b \text {arccosh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^4}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.72 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e \left (\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}-2 b (c+d x) \left (-4 a^3 (c+d x)-6 a b^2 (c+d x)+6 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+3 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+3 b^2 \left (-2 a^2-b^2+4 a^2 (c+d x)^2+2 b^2 (c+d x)^2-4 a b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+4 b^3 \left (-a+2 a (c+d x)^2-b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^3+b^4 \left (-1+2 (c+d x)^2\right ) \text {arccosh}(c+d x)^4-2 a b \left (2 a^2+3 b^2\right ) \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{4 d} \]

[In]

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

[Out]

(e*((2*a^4 + 6*a^2*b^2 + 3*b^4)*(c + d*x)^2 - 2*a*b*(2*a^2 + 3*b^2)*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c +
d*x] - 2*b*(c + d*x)*(-4*a^3*(c + d*x) - 6*a*b^2*(c + d*x) + 6*a^2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + 3*
b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + 3*b^2*(-2*a^2 - b^2 + 4*a^2*(c + d*x)^2 + 2*b^2*(
c + d*x)^2 - 4*a*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 + 4*b^3*(-a + 2*a*(c + d
*x)^2 - b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^3 + b^4*(-1 + 2*(c + d*x)^2)*ArcCos
h[c + d*x]^4 - 2*a*b*(2*a^2 + 3*b^2)*Log[c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]]))/(4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(421\) vs. \(2(189)=378\).

Time = 0.10 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.02

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\frac {\operatorname {arccosh}\left (d x +c \right )^{4}}{4}+\frac {3 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}\right )+4 e a \,b^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3}}{2}-\frac {3 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4}-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {3 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )+4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(422\)
default	\(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\frac {\operatorname {arccosh}\left (d x +c \right )^{4}}{4}+\frac {3 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}\right )+4 e a \,b^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3}}{2}-\frac {3 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4}-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {3 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )+4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(422\)
parts	\(e \,a^{4} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{4} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\frac {\operatorname {arccosh}\left (d x +c \right )^{4}}{4}+\frac {3 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}\right )}{d}+\frac {4 e a \,b^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3}}{2}-\frac {3 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4}-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {3 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right )}{8}\right )}{d}+\frac {6 e \,a^{2} b^{2} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )}{d}+\frac {4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(432\)

[In]

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

[Out]

1/d*(1/2*e*a^4*(d*x+c)^2+e*b^4*(1/2*(d*x+c)^2*arccosh(d*x+c)^4-(d*x+c)*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c
+1)^(1/2)-1/4*arccosh(d*x+c)^4+3/2*(d*x+c)^2*arccosh(d*x+c)^2-3/2*(d*x+c)*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+
c+1)^(1/2)-3/4*arccosh(d*x+c)^2+3/4*(d*x+c)^2)+4*e*a*b^3*(1/2*(d*x+c)^2*arccosh(d*x+c)^3-3/4*(d*x+c)*arccosh(d
*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-1/4*arccosh(d*x+c)^3+3/4*(d*x+c)^2*arccosh(d*x+c)-3/8*(d*x+c+1)^(1/2)*
(d*x+c-1)^(1/2)*(d*x+c)-3/8*arccosh(d*x+c))+6*e*a^2*b^2*(1/2*(d*x+c)^2*arccosh(d*x+c)^2-1/2*(d*x+c)*arccosh(d*
x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-1/4*arccosh(d*x+c)^2+1/4*(d*x+c)^2)+4*e*b*a^3*(1/2*(d*x+c)^2*arccosh(d*x+
c)-1/4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*((d*x+c)*((d*x+c)^2-1)^(1/2)+ln(d*x+c+((d*x+c)^2-1)^(1/2)))/((d*x+c)^2-
1)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (189) = 378\).

Time = 0.28 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.77 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} - b^{4}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} - a b^{3}\right )} e - {\left (b^{4} d e x + b^{4} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x - {\left (2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x - {\left (2 \, a^{3} b + 3 \, a b^{3} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \, {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]

[In]

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

[Out]

1/4*((2*a^4 + 6*a^2*b^2 + 3*b^4)*d^2*e*x^2 + 2*(2*a^4 + 6*a^2*b^2 + 3*b^4)*c*d*e*x + (2*b^4*d^2*e*x^2 + 4*b^4*
c*d*e*x + (2*b^4*c^2 - b^4)*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^4 + 4*(2*a*b^3*d^2*e*x^2 + 4*a
*b^3*c*d*e*x + (2*a*b^3*c^2 - a*b^3)*e - (b^4*d*e*x + b^4*c*e)*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 + 3*(2*(2*a^2*b^2 + b^4)*d^2*e*x^2 + 4*(2*a^2*b^2 + b^4)*c*d*e*x - (2*a
^2*b^2 + b^4 - 2*(2*a^2*b^2 + b^4)*c^2)*e - 4*(a*b^3*d*e*x + a*b^3*c*e)*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 + 2*(2*(2*a^3*b + 3*a*b^3)*d^2*e*x^2 + 4*(2*a^3*b + 3*a*b^3)*c
*d*e*x - (2*a^3*b + 3*a*b^3 - 2*(2*a^3*b + 3*a*b^3)*c^2)*e - 3*((2*a^2*b^2 + b^4)*d*e*x + (2*a^2*b^2 + b^4)*c*
e)*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*((2*a^3*b + 3*a*b^3
)*d*e*x + (2*a^3*b + 3*a*b^3)*c*e)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*a^4*d*e*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))*a^3*b*d*e + a^4*c*e*x + 4*((d*x + c)*arccosh(d*
x + c) - sqrt((d*x + c)^2 - 1))*a^3*b*c*e/d + 1/2*(b^4*d*e*x^2 + 2*b^4*c*e*x)*log(d*x + sqrt(d*x + c + 1)*sqrt
(d*x + c - 1) + c)^4 + integrate(2*((2*(c^4*e - c^2*e)*a*b^3 + (2*a*b^3*d^4*e - b^4*d^4*e)*x^4 + 4*(2*a*b^3*c*
d^3*e - b^4*c*d^3*e)*x^3 + (2*(6*c^2*d^2*e - d^2*e)*a*b^3 - (5*c^2*d^2*e - d^2*e)*b^4)*x^2 + (2*(c^3*e - c*e)*
a*b^3 + (2*a*b^3*d^3*e - b^4*d^3*e)*x^3 + 3*(2*a*b^3*c*d^2*e - b^4*c*d^2*e)*x^2 - 2*(b^4*c^2*d*e - (3*c^2*d*e
- d*e)*a*b^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(2*(2*c^3*d*e - c*d*e)*a*b^3 - (c^3*d*e - c*d*e)*b^4)
*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + 3*(a^2*b^2*d^4*e*x^4 + 4*a^2*b^2*c*d^3*e*x^3 + (6*c
^2*d^2*e - d^2*e)*a^2*b^2*x^2 + 2*(2*c^3*d*e - c*d*e)*a^2*b^2*x + (c^4*e - c^2*e)*a^2*b^2 + (a^2*b^2*d^3*e*x^3
+ 3*a^2*b^2*c*d^2*e*x^2 + (3*c^2*d*e - d*e)*a^2*b^2*x + (c^3*e - c*e)*a^2*b^2)*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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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