\(\int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx\) [123]

Optimal result

Integrand size = 23, antiderivative size = 309 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}-\frac {8 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}+\frac {8 b^2 e^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}-\frac {8 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d} \]

[Out]

160/27*b^4*e^2*x+8/81*b^4*e^2*(d*x+c)^3/d+8/3*b^2*e^2*(d*x+c)*(a+b*arccosh(d*x+c))^2/d+4/9*b^2*e^2*(d*x+c)^3*(
a+b*arccosh(d*x+c))^2/d+1/3*e^2*(d*x+c)^3*(a+b*arccosh(d*x+c))^4/d-160/27*b^3*e^2*(a+b*arccosh(d*x+c))*(d*x+c-
1)^(1/2)*(d*x+c+1)^(1/2)/d-8/27*b^3*e^2*(d*x+c)^2*(a+b*arccosh(d*x+c))*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-8/9*b
*e^2*(a+b*arccosh(d*x+c))^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d-4/9*b*e^2*(d*x+c)^2*(a+b*arccosh(d*x+c))^3*(d*x+
c-1)^(1/2)*(d*x+c+1)^(1/2)/d

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5996, 12, 5883, 5939, 5915, 5879, 8, 30} \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=-\frac {8 b^3 e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{27 d}-\frac {160 b^3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{27 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}+\frac {8 b^2 e^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{3 d}-\frac {4 b e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {8 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d}+\frac {8 b^4 e^2 (c+d x)^3}{81 d}+\frac {160}{27} b^4 e^2 x \]

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(160*b^4*e^2*x)/27 + (8*b^4*e^2*(c + d*x)^3)/(81*d) - (160*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b
*ArcCosh[c + d*x]))/(27*d) - (8*b^3*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*
x]))/(27*d) + (8*b^2*e^2*(c + d*x)*(a + b*ArcCosh[c + d*x])^2)/(3*d) + (4*b^2*e^2*(c + d*x)^3*(a + b*ArcCosh[c
+ d*x])^2)/(9*d) - (8*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(9*d) - (4*b*e^2
*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(9*d) + (e^2*(c + d*x)^3*(a + b*
ArcCosh[c + d*x])^4)/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 30

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

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(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, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -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^2 x^2 (a+b \text {arccosh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \text {arccosh}(x))^4 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d}-\frac {\left (4 b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arccosh}(x))^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d}-\frac {\left (8 b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \text {arccosh}(x))^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}+\frac {\left (4 b^2 e^2\right ) \text {Subst}\left (\int x^2 (a+b \text {arccosh}(x))^2 \, dx,x,c+d x\right )}{3 d} \\ & = \frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}-\frac {8 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d}+\frac {\left (8 b^2 e^2\right ) \text {Subst}\left (\int (a+b \text {arccosh}(x))^2 \, dx,x,c+d x\right )}{3 d}-\frac {\left (8 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \text {arccosh}(x))}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {8 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}+\frac {8 b^2 e^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}-\frac {8 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d}-\frac {\left (16 b^3 e^2\right ) \text {Subst}\left (\int \frac {x (a+b \text {arccosh}(x))}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{27 d}-\frac {\left (16 b^3 e^2\right ) \text {Subst}\left (\int \frac {x (a+b \text {arccosh}(x))}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (8 b^4 e^2\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{27 d} \\ & = \frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}-\frac {8 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}+\frac {8 b^2 e^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}-\frac {8 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d}+\frac {\left (16 b^4 e^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{27 d}+\frac {\left (16 b^4 e^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{3 d} \\ & = \frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}-\frac {8 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{27 d}+\frac {8 b^2 e^2 (c+d x) (a+b \text {arccosh}(c+d x))^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^2}{9 d}-\frac {8 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}-\frac {4 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.54 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e^2 \left (24 b^2 \left (9 a^2+20 b^2\right ) (c+d x)+\left (27 a^4+36 a^2 b^2+8 b^4\right ) (c+d x)^3+12 a b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-6 a^2-40 b^2-\left (3 a^2+2 b^2\right ) (c+d x)^2\right )+12 b \left (36 a b^2 (c+d x)+9 a^3 (c+d x)^3+6 a b^2 (c+d x)^3-18 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-40 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}-9 a^2 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}-2 b^3 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+18 b^2 \left (12 b^2 (c+d x)+9 a^2 (c+d x)^3+2 b^2 (c+d x)^3-12 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}-6 a b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2-36 b^3 \left (-3 a (c+d x)^3+2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}+b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^3+27 b^4 (c+d x)^3 \text {arccosh}(c+d x)^4\right )}{81 d} \]

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(e^2*(24*b^2*(9*a^2 + 20*b^2)*(c + d*x) + (27*a^4 + 36*a^2*b^2 + 8*b^4)*(c + d*x)^3 + 12*a*b*Sqrt[-1 + c + d*x
]*Sqrt[1 + c + d*x]*(-6*a^2 - 40*b^2 - (3*a^2 + 2*b^2)*(c + d*x)^2) + 12*b*(36*a*b^2*(c + d*x) + 9*a^3*(c + d*
x)^3 + 6*a*b^2*(c + d*x)^3 - 18*a^2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 40*b^3*Sqrt[-1 + c + d*x]*Sqrt[1
+ c + d*x] - 9*a^2*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] - 2*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^2*S
qrt[1 + c + d*x])*ArcCosh[c + d*x] + 18*b^2*(12*b^2*(c + d*x) + 9*a^2*(c + d*x)^3 + 2*b^2*(c + d*x)^3 - 12*a*b
*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 6*a*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*
x]^2 - 36*b^3*(-3*a*(c + d*x)^3 + 2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*Sqrt[-1 + c + d*x]*(c + d*x)^2*
Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^3 + 27*b^4*(c + d*x)^3*ArcCosh[c + d*x]^4))/(81*d)

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{4}}{3}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {8 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {160 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {160 d x}{27}+\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{9}-\frac {8 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+4 e^{2} b \,a^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\)	\(517\)
default	\(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{4}}{3}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {8 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {160 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {160 d x}{27}+\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{9}-\frac {8 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+4 e^{2} b \,a^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\)	\(517\)
parts	\(\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{4}}{3}-\frac {8 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {8 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {160 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {160 d x}{27}+\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{9}-\frac {8 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )}{d}+\frac {4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3}}{3}-\frac {2 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3}+\frac {4 \left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {40 \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{27}\right )}{d}+\frac {6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2}}{3}-\frac {4 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}-\frac {2 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{9}+\frac {4 d x}{9}+\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {4 e^{2} b \,a^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\)	\(528\)

[In]

int((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*e^2*a^4*(d*x+c)^3+e^2*b^4*(1/3*(d*x+c)^3*arccosh(d*x+c)^4-8/9*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c
+1)^(1/2)-4/9*(d*x+c)^2*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+8/3*(d*x+c)*arccosh(d*x+c)^2-160/27*a
rccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+160/27*d*x+160/27*c+4/9*(d*x+c)^3*arccosh(d*x+c)^2-8/27*(d*x+c)^
2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+8/81*(d*x+c)^3)+4*e^2*a*b^3*(1/3*(d*x+c)^3*arccosh(d*x+c)^3-2
/3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-1/3*(d*x+c)^2*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(
1/2)+4/3*(d*x+c)*arccosh(d*x+c)-40/27*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+2/9*(d*x+c)^3*arccosh(d*x+c)-2/27*(d*x+c
)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))+6*e^2*a^2*b^2*(1/3*(d*x+c)^3*arccosh(d*x+c)^2-4/9*arccosh(d*x+c)*(d*x+c-1
)^(1/2)*(d*x+c+1)^(1/2)-2/9*(d*x+c)^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4/9*d*x+4/9*c+2/27*(d*x+c
)^3)+4*e^2*b*a^3*(1/3*(d*x+c)^3*arccosh(d*x+c)-1/9*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*((d*x+c)^2+2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (275) = 550\).

Time = 0.29 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.88 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {{\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left (72 \, a^{2} b^{2} + 160 \, b^{4} + {\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c^{2}\right )} d e^{2} x + 27 \, {\left (b^{4} d^{3} e^{2} x^{3} + 3 \, b^{4} c d^{2} e^{2} x^{2} + 3 \, b^{4} c^{2} d e^{2} x + b^{4} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{4} + 36 \, {\left (3 \, a b^{3} d^{3} e^{2} x^{3} + 9 \, a b^{3} c d^{2} e^{2} x^{2} + 9 \, a b^{3} c^{2} d e^{2} x + 3 \, a b^{3} c^{3} e^{2} - {\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + {\left (b^{4} c^{2} + 2 \, b^{4}\right )} e^{2}\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} + 18 \, {\left ({\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left (4 \, b^{4} + {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{2}\right )} d e^{2} x + {\left (12 \, b^{4} c + {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{3}\right )} e^{2} - 6 \, {\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + {\left (a b^{3} c^{2} + 2 \, a b^{3}\right )} e^{2}\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} + 12 \, {\left (3 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c d^{2} e^{2} x^{2} + 9 \, {\left (4 \, a b^{3} + {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{2}\right )} d e^{2} x + 3 \, {\left (12 \, a b^{3} c + {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{3}\right )} e^{2} - {\left ({\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c d e^{2} x + {\left (18 \, a^{2} b^{2} + 40 \, b^{4} + {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{2}\right )} e^{2}\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 ) - 12 \, {\left ({\left (3 \, a^{3} b + 2 \, a b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c d e^{2} x + {\left (6 \, a^{3} b + 40 \, a b^{3} + {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{81 \, d} \]

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/81*((27*a^4 + 36*a^2*b^2 + 8*b^4)*d^3*e^2*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 8*b^4)*c*d^2*e^2*x^2 + 3*(72*a^2*b^
2 + 160*b^4 + (27*a^4 + 36*a^2*b^2 + 8*b^4)*c^2)*d*e^2*x + 27*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4*c
^2*d*e^2*x + b^4*c^3*e^2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^4 + 36*(3*a*b^3*d^3*e^2*x^3 + 9*a*b
^3*c*d^2*e^2*x^2 + 9*a*b^3*c^2*d*e^2*x + 3*a*b^3*c^3*e^2 - (b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + (b^4*c^2 + 2*b
^4)*e^2)*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 + 18*((9*a^2*b^
2 + 2*b^4)*d^3*e^2*x^3 + 3*(9*a^2*b^2 + 2*b^4)*c*d^2*e^2*x^2 + 3*(4*b^4 + (9*a^2*b^2 + 2*b^4)*c^2)*d*e^2*x + (
12*b^4*c + (9*a^2*b^2 + 2*b^4)*c^3)*e^2 - 6*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2*x + (a*b^3*c^2 + 2*a*b^3)*e^2
)*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 + 12*(3*(3*a^3*b + 2*a
*b^3)*d^3*e^2*x^3 + 9*(3*a^3*b + 2*a*b^3)*c*d^2*e^2*x^2 + 9*(4*a*b^3 + (3*a^3*b + 2*a*b^3)*c^2)*d*e^2*x + 3*(1
2*a*b^3*c + (3*a^3*b + 2*a*b^3)*c^3)*e^2 - ((9*a^2*b^2 + 2*b^4)*d^2*e^2*x^2 + 2*(9*a^2*b^2 + 2*b^4)*c*d*e^2*x
+ (18*a^2*b^2 + 40*b^4 + (9*a^2*b^2 + 2*b^4)*c^2)*e^2)*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)) - 12*((3*a^3*b + 2*a*b^3)*d^2*e^2*x^2 + 2*(3*a^3*b + 2*a*b^3)*c*d*e^2*x + (6*a^3*
b + 40*a*b^3 + (3*a^3*b + 2*a*b^3)*c^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

Sympy [F]

\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=e^{2} \left (\int a^{4} c^{2}\, dx + \int a^{4} d^{2} x^{2}\, dx + \int b^{4} c^{2} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 4 a b^{3} c^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a^{2} b^{2} c^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a^{3} b c^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 a^{4} c d x\, dx + \int b^{4} d^{2} x^{2} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 4 a b^{3} d^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a^{2} b^{2} d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a^{3} b d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 b^{4} c d x \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 8 a b^{3} c d x \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 12 a^{2} b^{2} c d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 8 a^{3} b c d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**2*(a+b*acosh(d*x+c))**4,x)

[Out]

e**2*(Integral(a**4*c**2, x) + Integral(a**4*d**2*x**2, x) + Integral(b**4*c**2*acosh(c + d*x)**4, x) + Integr
al(4*a*b**3*c**2*acosh(c + d*x)**3, x) + Integral(6*a**2*b**2*c**2*acosh(c + d*x)**2, x) + Integral(4*a**3*b*c
**2*acosh(c + d*x), x) + Integral(2*a**4*c*d*x, x) + Integral(b**4*d**2*x**2*acosh(c + d*x)**4, x) + Integral(
4*a*b**3*d**2*x**2*acosh(c + d*x)**3, x) + Integral(6*a**2*b**2*d**2*x**2*acosh(c + d*x)**2, x) + Integral(4*a
**3*b*d**2*x**2*acosh(c + d*x), x) + Integral(2*b**4*c*d*x*acosh(c + d*x)**4, x) + Integral(8*a*b**3*c*d*x*aco
sh(c + d*x)**3, x) + Integral(12*a**2*b**2*c*d*x*acosh(c + d*x)**2, x) + Integral(8*a**3*b*c*d*x*acosh(c + d*x
), x))

Maxima [F]

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

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

1/3*a^4*d^2*e^2*x^3 + a^4*c*d*e^2*x^2 + 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*c*d*e^2 + 2/9*(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))*a^3*b*d^2*e^2 + a^4*c^2*e^2*x + 4*((d*x + c)*arccosh(d*x + c) - sqrt
((d*x + c)^2 - 1))*a^3*b*c^2*e^2/d + 1/3*(b^4*d^2*e^2*x^3 + 3*b^4*c*d*e^2*x^2 + 3*b^4*c^2*e^2*x)*log(d*x + sqr
t(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4 + integrate(2/3*(2*((3*a*b^3*d^5*e^2 - b^4*d^5*e^2)*x^5 + 3*(c^5*e^2 -
c^3*e^2)*a*b^3 + 5*(3*a*b^3*c*d^4*e^2 - b^4*c*d^4*e^2)*x^4 + (3*(10*c^2*d^3*e^2 - d^3*e^2)*a*b^3 - (10*c^2*d^
3*e^2 - d^3*e^2)*b^4)*x^3 + 3*((10*c^3*d^2*e^2 - 3*c*d^2*e^2)*a*b^3 - (3*c^3*d^2*e^2 - c*d^2*e^2)*b^4)*x^2 + (
3*(c^4*e^2 - c^2*e^2)*a*b^3 + (3*a*b^3*d^4*e^2 - b^4*d^4*e^2)*x^4 + 4*(3*a*b^3*c*d^3*e^2 - b^4*c*d^3*e^2)*x^3
- 3*(2*b^4*c^2*d^2*e^2 - (6*c^2*d^2*e^2 - d^2*e^2)*a*b^3)*x^2 - 3*(b^4*c^3*d*e^2 - 2*(2*c^3*d*e^2 - c*d*e^2)*a
*b^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 3*((5*c^4*d*e^2 - 3*c^2*d*e^2)*a*b^3 - (c^4*d*e^2 - c^2*d*e^2)*
b^4)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + 9*(a^2*b^2*d^5*e^2*x^5 + 5*a^2*b^2*c*d^4*e^2*x^
4 + (10*c^2*d^3*e^2 - d^3*e^2)*a^2*b^2*x^3 + (10*c^3*d^2*e^2 - 3*c*d^2*e^2)*a^2*b^2*x^2 + (5*c^4*d*e^2 - 3*c^2
*d*e^2)*a^2*b^2*x + (c^5*e^2 - c^3*e^2)*a^2*b^2 + (a^2*b^2*d^4*e^2*x^4 + 4*a^2*b^2*c*d^3*e^2*x^3 + (6*c^2*d^2*
e^2 - d^2*e^2)*a^2*b^2*x^2 + 2*(2*c^3*d*e^2 - c*d*e^2)*a^2*b^2*x + (c^4*e^2 - c^2*e^2)*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]

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

[In]

integrate((d*e*x+c*e)^2*(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2*(b*arccosh(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \]

[In]

int((c*e + d*e*x)^2*(a + b*acosh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^2*(a + b*acosh(c + d*x))^4, x)