3.122 \(\int (c e+d e x)^3 (a+b \cosh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=377 \[ -\frac{3 b^3 e^3 \sqrt{c+d x-1} (c+d x)^3 \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{45 b^3 e^3 \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{64 d}+\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{45 b^2 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{128 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{b e^3 \sqrt{c+d x-1} (c+d x)^3 \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{3 b e^3 \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{8 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{32 d}+\frac{3 b^4 e^3 (c+d x)^4}{128 d}+\frac{45 b^4 e^3 (c+d x)^2}{128 d} \]

[Out]

(45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) - (45*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*
Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(64*d) - (3*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*
x]*(a + b*ArcCosh[c + d*x]))/(32*d) - (45*b^2*e^3*(a + b*ArcCosh[c + d*x])^2)/(128*d) + (9*b^2*e^3*(c + d*x)^2
*(a + b*ArcCosh[c + d*x])^2)/(16*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^2)/(16*d) - (3*b*e^3*Sqr
t[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(8*d) - (b*e^3*Sqrt[-1 + c + d*x]*(c +
d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(4*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^4)/(32*d) + (e^3
*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^4)/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 1.17449, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 12, 5662, 5759, 5676, 30} \[ -\frac{3 b^3 e^3 \sqrt{c+d x-1} (c+d x)^3 \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{32 d}-\frac{45 b^3 e^3 \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{64 d}+\frac{3 b^2 e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{45 b^2 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{128 d}+\frac{9 b^2 e^3 (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{16 d}-\frac{b e^3 \sqrt{c+d x-1} (c+d x)^3 \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{3 b e^3 \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{8 d}+\frac{e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{32 d}+\frac{3 b^4 e^3 (c+d x)^4}{128 d}+\frac{45 b^4 e^3 (c+d x)^2}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) - (45*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*
Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(64*d) - (3*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*
x]*(a + b*ArcCosh[c + d*x]))/(32*d) - (45*b^2*e^3*(a + b*ArcCosh[c + d*x])^2)/(128*d) + (9*b^2*e^3*(c + d*x)^2
*(a + b*ArcCosh[c + d*x])^2)/(16*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^2)/(16*d) - (3*b*e^3*Sqr
t[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(8*d) - (b*e^3*Sqrt[-1 + c + d*x]*(c +
d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(4*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^4)/(32*d) + (e^3
*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^4)/(4*d)

Rule 5866

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

Rule 12

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

Rule 5662

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

Rule 5759

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

Rule 5676

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

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.806236, size = 562, normalized size = 1.49 \[ \frac{e^3 \left (\left (24 a^2 b^2+32 a^4+3 b^4\right ) (c+d x)^4+9 b^2 \left (8 a^2+5 b^2\right ) (c+d x)^2+2 a b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \left (-2 \left (8 a^2+3 b^2\right ) (c+d x)^2-3 \left (8 a^2+15 b^2\right )\right )-6 a b \left (8 a^2+15 b^2\right ) \log \left (\sqrt{c+d x-1} \sqrt{c+d x+1}+c+d x\right )+2 b (c+d x) \cosh ^{-1}(c+d x) \left (-48 a^2 b \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1}-72 a^2 b \sqrt{c+d x-1} \sqrt{c+d x+1}+64 a^3 (c+d x)^3+24 a b^2 (c+d x)^3+72 a b^2 (c+d x)-6 b^3 \sqrt{c+d x-1} (c+d x)^2 \sqrt{c+d x+1}-45 b^3 \sqrt{c+d x-1} \sqrt{c+d x+1}\right )+3 b^2 \cosh ^{-1}(c+d x)^2 \left (64 a^2 (c+d x)^4-24 a^2-32 a b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3-48 a b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)+8 b^2 (c+d x)^4+24 b^2 (c+d x)^2-15 b^2\right )+16 b^3 \cosh ^{-1}(c+d x)^3 \left (8 a (c+d x)^4-3 a-2 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^3-3 b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)\right )+4 b^4 \left (8 (c+d x)^4-3\right ) \cosh ^{-1}(c+d x)^4\right )}{128 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(e^3*(9*b^2*(8*a^2 + 5*b^2)*(c + d*x)^2 + (32*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x)^4 + 2*a*b*Sqrt[-1 + c + d*x]
*(c + d*x)*Sqrt[1 + c + d*x]*(-3*(8*a^2 + 15*b^2) - 2*(8*a^2 + 3*b^2)*(c + d*x)^2) + 2*b*(c + d*x)*(72*a*b^2*(
c + d*x) + 64*a^3*(c + d*x)^3 + 24*a*b^2*(c + d*x)^3 - 72*a^2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 45*b^3*
Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 48*a^2*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] - 6*b^3*Sqrt[
-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + 3*b^2*(-24*a^2 - 15*b^2 + 24*b^2*(c + d*x)^2 +
64*a^2*(c + d*x)^4 + 8*b^2*(c + d*x)^4 - 48*a*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x] - 32*a*b*Sqrt[
-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 + 16*b^3*(-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]^3 + 4*b^4*(-3 + 8*(c + d*x)^4)*ArcCosh[c + d*x]^4 - 6*a*b*(8*a^2 + 15*b^2)*Log[c + d*x + Sqrt[-1 + c + d*x]
*Sqrt[1 + c + d*x]]))/(128*d)

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 2465, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x)

[Out]

-9/32*d*e^3*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-3/4*d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*a^3*b*c*e^3-
3/8/d*e^3*a^3*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/((d*x+c)^2-1)^(1/2)*ln(d*x+c+((d*x+c)^2-1)^(1/2))-3/4/d*e^3*a*
b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-9/32*d*e^3*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+
1)^(1/2)*x^2*c-3/4*d*e^3*b^4*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-3/4/d*e^3*a^2*b^2*arccosh(
d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-9/8/d*e^3*a^2*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-
9/4*d*e^3*a*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-9/4*d*e^3*a^2*b^2*arccosh(d*x+c)*(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2)*x^2*c-9/32*e^3*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-3/32/d*e^3*a*b^3*(d*x+c-1
)^(1/2)*(d*x+c+1)^(1/2)*c^3-45/64/d*e^3*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-3/8/d*e^3*b^4*arccosh(d*x+c)^3
*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-3/32/d*e^3*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-45/64/d*e
^3*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-1/4/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*a^3*b*c^3*e^3-3/
8/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*a^3*b*c*e^3-1/4*d^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^3*a^3*b*e^3+4*d^2*ar
ccosh(d*x+c)*x^3*a^3*b*c*e^3+6*d*arccosh(d*x+c)*x^2*a^3*b*c^2*e^3-3/32*d^2*e^3*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)
^(1/2)*x^3+4*d^2*e^3*a*b^3*arccosh(d*x+c)^3*x^3*c+6*d*e^3*a*b^3*arccosh(d*x+c)^3*x^2*c^2-1/4*d^2*e^3*b^4*arcco
sh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^3+3/2*d^2*e^3*a*b^3*arccosh(d*x+c)*x^3*c+9/4*d*e^3*a*b^3*arccosh
(d*x+c)*x^2*c^2-3/4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*a^3*b*c^2*e^3+45/128/d*e^3*b^4*c^2+3/128/d*e^3*b^4*c^4+1
/4/d*a^4*c^4*e^3+3/128*d^3*e^3*b^4*x^4+45/128*d*e^3*b^4*x^2+1/4*d^3*x^4*a^4*e^3-45/128/d*e^3*b^4*arccosh(d*x+c
)^2-3/32/d*e^3*b^4*arccosh(d*x+c)^4+45/64*e^3*b^4*x*c+3/32*e^3*b^4*x*c^3+x*a^4*c^3*e^3+3/16*d^3*e^3*a^2*b^2*x^
4+9/16*d*e^3*a^2*b^2*x^2+d^2*x^3*a^4*c*e^3+3/32*d^2*e^3*b^4*x^3*c+9/64*d*e^3*b^4*x^2*c^2+3/2*d*x^2*a^4*c^2*e^3
+3/16*d^3*e^3*b^4*arccosh(d*x+c)^2*x^4+9/16*d*e^3*b^4*arccosh(d*x+c)^2*x^2+1/4*d^3*e^3*b^4*arccosh(d*x+c)^4*x^
4+e^3*b^4*arccosh(d*x+c)^4*x*c^3+3/4*e^3*b^4*arccosh(d*x+c)^2*x*c^3+9/8*e^3*b^4*arccosh(d*x+c)^2*x*c-9/16/d*e^
3*a^2*b^2*arccosh(d*x+c)^2-45/64/d*e^3*a*b^3*arccosh(d*x+c)-3/8/d*e^3*a*b^3*arccosh(d*x+c)^3+9/16/d*e^3*b^4*ar
ccosh(d*x+c)^2*c^2+1/4/d*e^3*b^4*arccosh(d*x+c)^4*c^4+3/4*e^3*a^2*b^2*x*c^3+9/8*e^3*a^2*b^2*x*c+d^3*e^3*a*b^3*
arccosh(d*x+c)^3*x^4+3/8*d^3*e^3*a*b^3*arccosh(d*x+c)*x^4+9/8*d*e^3*a*b^3*arccosh(d*x+c)*x^2+3/2*d^3*e^3*a^2*b
^2*arccosh(d*x+c)^2*x^4-45/64*e^3*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x+4*e^3*a*b^3*arccosh(d*x+c)^3*x*c^3-3
/8*e^3*b^4*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x+3/2*d*e^3*b^4*arccosh(d*x+c)^4*x^2*c^2+3/4*d^2*e
^3*b^4*arccosh(d*x+c)^2*x^3*c+9/8*d*e^3*b^4*arccosh(d*x+c)^2*x^2*c^2+3/2/d*e^3*a^2*b^2*arccosh(d*x+c)^2*c^4+1/
d*arccosh(d*x+c)*a^3*b*c^4*e^3+1/d*e^3*a*b^3*arccosh(d*x+c)^3*c^4+d^2*e^3*b^4*arccosh(d*x+c)^4*x^3*c-3/8*(d*x+
c-1)^(1/2)*(d*x+c+1)^(1/2)*x*a^3*b*e^3+4*arccosh(d*x+c)*x*a^3*b*c^3*e^3+3/4*d^2*e^3*a^2*b^2*x^3*c+9/8*d*e^3*a^
2*b^2*x^2*c^2+3/2*e^3*a*b^3*arccosh(d*x+c)*x*c^3+9/4*e^3*a*b^3*arccosh(d*x+c)*x*c-45/64*e^3*b^4*arccosh(d*x+c)
*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x+6*e^3*a^2*b^2*arccosh(d*x+c)^2*x*c^3+3/8/d*e^3*a*b^3*arccosh(d*x+c)*c^4+9/8
/d*e^3*a*b^3*arccosh(d*x+c)*c^2+d^3*arccosh(d*x+c)*x^4*a^3*b*e^3+9/16/d*e^3*a^2*b^2*c^2+3/16/d*e^3*a^2*b^2*c^4
+3/16/d*e^3*b^4*arccosh(d*x+c)^2*c^4-9/32*e^3*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-3/4*e^3
*b^4*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-3/32*d^2*e^3*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d
*x+c+1)^(1/2)*x^3+6*d^2*e^3*a^2*b^2*arccosh(d*x+c)^2*x^3*c+9*d*e^3*a^2*b^2*arccosh(d*x+c)^2*x^2*c^2-1/4/d*e^3*
b^4*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c^3-9/8*e^3*a*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c
+1)^(1/2)*x-9/8*e^3*a^2*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x-3/4*d^2*e^3*a*b^3*arccosh(d*x+c)^
2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^3-9/4*e^3*a*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-9/4
*e^3*a^2*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*c^2-9/8/d*e^3*a*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(
1/2)*(d*x+c+1)^(1/2)*c-3/4*d^2*e^3*a^2*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x^3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.87925, size = 2631, normalized size = 6.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/128*((32*a^4 + 24*a^2*b^2 + 3*b^4)*d^4*e^3*x^4 + 4*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c*d^3*e^3*x^3 + 3*(24*a^2*b
^2 + 15*b^4 + 2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^2)*d^2*e^3*x^2 + 2*(2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^3 + 9*(8
*a^2*b^2 + 5*b^4)*c)*d*e^3*x + 4*(8*b^4*d^4*e^3*x^4 + 32*b^4*c*d^3*e^3*x^3 + 48*b^4*c^2*d^2*e^3*x^2 + 32*b^4*c
^3*d*e^3*x + (8*b^4*c^4 - 3*b^4)*e^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^4 + 16*(8*a*b^3*d^4*e^3
*x^4 + 32*a*b^3*c*d^3*e^3*x^3 + 48*a*b^3*c^2*d^2*e^3*x^2 + 32*a*b^3*c^3*d*e^3*x + (8*a*b^3*c^4 - 3*a*b^3)*e^3
- (2*b^4*d^3*e^3*x^3 + 6*b^4*c*d^2*e^3*x^2 + 3*(2*b^4*c^2 + b^4)*d*e^3*x + (2*b^4*c^3 + 3*b^4*c)*e^3)*sqrt(d^2
*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^3 + 3*(8*(8*a^2*b^2 + b^4)*d^4*e^3
*x^4 + 32*(8*a^2*b^2 + b^4)*c*d^3*e^3*x^3 + 24*(b^4 + 2*(8*a^2*b^2 + b^4)*c^2)*d^2*e^3*x^2 + 16*(3*b^4*c + 2*(
8*a^2*b^2 + b^4)*c^3)*d*e^3*x + (24*b^4*c^2 + 8*(8*a^2*b^2 + b^4)*c^4 - 24*a^2*b^2 - 15*b^4)*e^3 - 16*(2*a*b^3
*d^3*e^3*x^3 + 6*a*b^3*c*d^2*e^3*x^2 + 3*(2*a*b^3*c^2 + a*b^3)*d*e^3*x + (2*a*b^3*c^3 + 3*a*b^3*c)*e^3)*sqrt(d
^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 2*(8*(8*a^3*b + 3*a*b^3)*d^4
*e^3*x^4 + 32*(8*a^3*b + 3*a*b^3)*c*d^3*e^3*x^3 + 24*(3*a*b^3 + 2*(8*a^3*b + 3*a*b^3)*c^2)*d^2*e^3*x^2 + 16*(9
*a*b^3*c + 2*(8*a^3*b + 3*a*b^3)*c^3)*d*e^3*x + (72*a*b^3*c^2 + 8*(8*a^3*b + 3*a*b^3)*c^4 - 24*a^3*b - 45*a*b^
3)*e^3 - 3*(2*(8*a^2*b^2 + b^4)*d^3*e^3*x^3 + 6*(8*a^2*b^2 + b^4)*c*d^2*e^3*x^2 + 3*(8*a^2*b^2 + 5*b^4 + 2*(8*
a^2*b^2 + b^4)*c^2)*d*e^3*x + (2*(8*a^2*b^2 + b^4)*c^3 + 3*(8*a^2*b^2 + 5*b^4)*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x
+ c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 2*(2*(8*a^3*b + 3*a*b^3)*d^3*e^3*x^3 + 6*(8*a^3
*b + 3*a*b^3)*c*d^2*e^3*x^2 + 3*(8*a^3*b + 15*a*b^3 + 2*(8*a^3*b + 3*a*b^3)*c^2)*d*e^3*x + (2*(8*a^3*b + 3*a*b
^3)*c^3 + 3*(8*a^3*b + 15*a*b^3)*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/d

________________________________________________________________________________________

Sympy [A]  time = 27.3642, size = 2876, normalized size = 7.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3*(a+b*acosh(d*x+c))**4,x)

[Out]

Piecewise((a**4*c**3*e**3*x + 3*a**4*c**2*d*e**3*x**2/2 + a**4*c*d**2*e**3*x**3 + a**4*d**3*e**3*x**4/4 + a**3
*b*c**4*e**3*acosh(c + d*x)/d + 4*a**3*b*c**3*e**3*x*acosh(c + d*x) - a**3*b*c**3*e**3*sqrt(c**2 + 2*c*d*x + d
**2*x**2 - 1)/(4*d) + 6*a**3*b*c**2*d*e**3*x**2*acosh(c + d*x) - 3*a**3*b*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d*
*2*x**2 - 1)/4 + 4*a**3*b*c*d**2*e**3*x**3*acosh(c + d*x) - 3*a**3*b*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)/4 - 3*a**3*b*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(8*d) + a**3*b*d**3*e**3*x**4*acosh(c + d*x
) - a**3*b*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/4 - 3*a**3*b*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)/8 - 3*a**3*b*e**3*acosh(c + d*x)/(8*d) + 3*a**2*b**2*c**4*e**3*acosh(c + d*x)**2/(2*d) + 6*a**2*b**2
*c**3*e**3*x*acosh(c + d*x)**2 + 3*a**2*b**2*c**3*e**3*x/4 - 3*a**2*b**2*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)*acosh(c + d*x)/(4*d) + 9*a**2*b**2*c**2*d*e**3*x**2*acosh(c + d*x)**2 + 9*a**2*b**2*c**2*d*e**3*x**2
/8 - 9*a**2*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/4 + 6*a**2*b**2*c*d**2*e**3*x
**3*acosh(c + d*x)**2 + 3*a**2*b**2*c*d**2*e**3*x**3/4 - 9*a**2*b**2*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 - 1)*acosh(c + d*x)/4 + 9*a**2*b**2*c*e**3*x/8 - 9*a**2*b**2*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*
acosh(c + d*x)/(8*d) + 3*a**2*b**2*d**3*e**3*x**4*acosh(c + d*x)**2/2 + 3*a**2*b**2*d**3*e**3*x**4/16 - 3*a**2
*b**2*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/4 + 9*a**2*b**2*d*e**3*x**2/16 - 9*a*
*2*b**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/8 - 9*a**2*b**2*e**3*acosh(c + d*x)**2/(16*
d) + a*b**3*c**4*e**3*acosh(c + d*x)**3/d + 3*a*b**3*c**4*e**3*acosh(c + d*x)/(8*d) + 4*a*b**3*c**3*e**3*x*aco
sh(c + d*x)**3 + 3*a*b**3*c**3*e**3*x*acosh(c + d*x)/2 - 3*a*b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 -
1)*acosh(c + d*x)**2/(4*d) - 3*a*b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(32*d) + 6*a*b**3*c**2*d*
e**3*x**2*acosh(c + d*x)**3 + 9*a*b**3*c**2*d*e**3*x**2*acosh(c + d*x)/4 - 9*a*b**3*c**2*e**3*x*sqrt(c**2 + 2*
c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/4 - 9*a*b**3*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/32 + 9*
a*b**3*c**2*e**3*acosh(c + d*x)/(8*d) + 4*a*b**3*c*d**2*e**3*x**3*acosh(c + d*x)**3 + 3*a*b**3*c*d**2*e**3*x**
3*acosh(c + d*x)/2 - 9*a*b**3*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/4 - 9*a*b**
3*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/32 + 9*a*b**3*c*e**3*x*acosh(c + d*x)/4 - 9*a*b**3*c*e**3
*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/(8*d) - 45*a*b**3*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x*
*2 - 1)/(64*d) + a*b**3*d**3*e**3*x**4*acosh(c + d*x)**3 + 3*a*b**3*d**3*e**3*x**4*acosh(c + d*x)/8 - 3*a*b**3
*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/4 - 3*a*b**3*d**2*e**3*x**3*sqrt(c**2 +
2*c*d*x + d**2*x**2 - 1)/32 + 9*a*b**3*d*e**3*x**2*acosh(c + d*x)/8 - 9*a*b**3*e**3*x*sqrt(c**2 + 2*c*d*x + d
**2*x**2 - 1)*acosh(c + d*x)**2/8 - 45*a*b**3*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/64 - 3*a*b**3*e**3*a
cosh(c + d*x)**3/(8*d) - 45*a*b**3*e**3*acosh(c + d*x)/(64*d) + b**4*c**4*e**3*acosh(c + d*x)**4/(4*d) + 3*b**
4*c**4*e**3*acosh(c + d*x)**2/(16*d) + b**4*c**3*e**3*x*acosh(c + d*x)**4 + 3*b**4*c**3*e**3*x*acosh(c + d*x)*
*2/4 + 3*b**4*c**3*e**3*x/32 - b**4*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**3/(4*d) - 3
*b**4*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(32*d) + 3*b**4*c**2*d*e**3*x**2*acosh(c +
d*x)**4/2 + 9*b**4*c**2*d*e**3*x**2*acosh(c + d*x)**2/8 + 9*b**4*c**2*d*e**3*x**2/64 - 3*b**4*c**2*e**3*x*sqr
t(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**3/4 - 9*b**4*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1
)*acosh(c + d*x)/32 + 9*b**4*c**2*e**3*acosh(c + d*x)**2/(16*d) + b**4*c*d**2*e**3*x**3*acosh(c + d*x)**4 + 3*
b**4*c*d**2*e**3*x**3*acosh(c + d*x)**2/4 + 3*b**4*c*d**2*e**3*x**3/32 - 3*b**4*c*d*e**3*x**2*sqrt(c**2 + 2*c*
d*x + d**2*x**2 - 1)*acosh(c + d*x)**3/4 - 9*b**4*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c +
d*x)/32 + 9*b**4*c*e**3*x*acosh(c + d*x)**2/8 + 45*b**4*c*e**3*x/64 - 3*b**4*c*e**3*sqrt(c**2 + 2*c*d*x + d**
2*x**2 - 1)*acosh(c + d*x)**3/(8*d) - 45*b**4*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(64*d
) + b**4*d**3*e**3*x**4*acosh(c + d*x)**4/4 + 3*b**4*d**3*e**3*x**4*acosh(c + d*x)**2/16 + 3*b**4*d**3*e**3*x*
*4/128 - b**4*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**3/4 - 3*b**4*d**2*e**3*x**3*
sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/32 + 9*b**4*d*e**3*x**2*acosh(c + d*x)**2/16 + 45*b**4*d*e
**3*x**2/128 - 3*b**4*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**3/8 - 45*b**4*e**3*x*sqrt(c*
*2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/64 - 3*b**4*e**3*acosh(c + d*x)**4/(32*d) - 45*b**4*e**3*acosh(c
+ d*x)**2/(128*d), Ne(d, 0)), (c**3*e**3*x*(a + b*acosh(c))**4, True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{3}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^4, x)