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

Optimal. Leaf size=209 \[ -\frac{3 b^3 e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}+\frac{3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac{3 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac{b e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d} \]

[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)

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Rubi [A]  time = 0.562042, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5866, 12, 5662, 5759, 5676, 30} \[ -\frac{3 b^3 e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}+\frac{3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac{3 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}-\frac{b e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d} \]

Antiderivative was successfully verified.

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

Mathematica [A]  time = 0.468614, size = 360, normalized size = 1.72 \[ \frac{e \left (\left (6 a^2 b^2+2 a^4+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}-2 a b \left (2 a^2+3 b^2\right ) \log \left (\sqrt{c+d x-1} \sqrt{c+d x+1}+c+d x\right )+3 b^2 \cosh ^{-1}(c+d x)^2 \left (4 a^2 (c+d x)^2-2 a^2-4 a b \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}+2 b^2 (c+d x)^2-b^2\right )-2 b (c+d x) \cosh ^{-1}(c+d x) \left (6 a^2 b \sqrt{c+d x-1} \sqrt{c+d x+1}-4 a^3 (c+d x)-6 a b^2 (c+d x)+3 b^3 \sqrt{c+d x-1} \sqrt{c+d x+1}\right )+4 b^3 \cosh ^{-1}(c+d x)^3 \left (2 a (c+d x)^2-a-b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)\right )+b^4 \left (2 (c+d x)^2-1\right ) \cosh ^{-1}(c+d x)^4\right )}{4 d} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.04, size = 933, normalized size = 4.5 \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)*(a+b*arccosh(d*x+c))^4,x)

[Out]

-1/d*e*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/d*e*a^2*b^2*a
rccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-3/d*e*a*b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c
-3/2/d*e*a^2*b^2*arccosh(d*x+c)^2+3*e*a^2*b^2*x*c-1/4/d*e*b^4*arccosh(d*x+c)^4-3/4/d*e*b^4*arccosh(d*x+c)^2+3/
2/d*e*a^2*b^2*c^2+3/2*d*e*a^2*b^2*x^2+e*b^4*arccosh(d*x+c)^4*x*c+3*e*b^4*arccosh(d*x+c)^2*x*c+1/2/d*e*b^4*arcc
osh(d*x+c)^4*c^2+3/2/d*e*b^4*arccosh(d*x+c)^2*c^2-1/d*e*a*b^3*arccosh(d*x+c)^3-3/2/d*e*a*b^3*arccosh(d*x+c)+1/
2*d*e*b^4*arccosh(d*x+c)^4*x^2+3/2*d*e*b^4*arccosh(d*x+c)^2*x^2+3/4/d*e*b^4*c^2+1/2/d*a^4*c^2*e+x*a^4*c*e+3/2*
e*b^4*x*c+1/2*d*x^2*a^4*e+3/4*d*e*b^4*x^2+2/d*e*a*b^3*arccosh(d*x+c)^3*c^2+3/d*e*a*b^3*arccosh(d*x+c)*c^2+3/d*
e*a^2*b^2*arccosh(d*x+c)^2*c^2+2/d*arccosh(d*x+c)*a^3*b*c^2*e+2*d*e*a*b^3*arccosh(d*x+c)^3*x^2+3*d*e*a*b^3*arc
cosh(d*x+c)*x^2+3*d*e*a^2*b^2*arccosh(d*x+c)^2*x^2-3/2*e*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x+
4*e*a*b^3*arccosh(d*x+c)^3*x*c+6*e*a*b^3*arccosh(d*x+c)*x*c-3/2*e*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x-(d*x
+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*a^3*b*e+4*arccosh(d*x+c)*x*a^3*b*c*e+6*e*a^2*b^2*arccosh(d*x+c)^2*x*c-e*b^4*arcc
osh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x-3*e*a^2*b^2*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x-3/
2/d*e*a*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-3/2/d*e*b^4*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-1/d
*e*b^4*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*c-1/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*a^3*b*c*e-3*e*a*
b^3*arccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x+2*d*arccosh(d*x+c)*x^2*a^3*b*e

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.61011, size = 1287, normalized size = 6.16 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 6.27775, size = 1027, normalized size = 4.91 \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)*(a+b*acosh(d*x+c))**4,x)

[Out]

Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*acosh(c + d*x)/d + 4*a**3*b*c*e*x*acosh(c + d*x) - a
**3*b*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/d + 2*a**3*b*d*e*x**2*acosh(c + d*x) - a**3*b*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 - 1) - a**3*b*e*acosh(c + d*x)/d + 3*a**2*b**2*c**2*e*acosh(c + d*x)**2/d + 6*a**2*b**2*c*e
*x*acosh(c + d*x)**2 + 3*a**2*b**2*c*e*x - 3*a**2*b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)
/d + 3*a**2*b**2*d*e*x**2*acosh(c + d*x)**2 + 3*a**2*b**2*d*e*x**2/2 - 3*a**2*b**2*e*x*sqrt(c**2 + 2*c*d*x + d
**2*x**2 - 1)*acosh(c + d*x) - 3*a**2*b**2*e*acosh(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*acosh(c + d*x)**3/d + 3
*a*b**3*c**2*e*acosh(c + d*x)/d + 4*a*b**3*c*e*x*acosh(c + d*x)**3 + 6*a*b**3*c*e*x*acosh(c + d*x) - 3*a*b**3*
c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 -
1)/(2*d) + 2*a*b**3*d*e*x**2*acosh(c + d*x)**3 + 3*a*b**3*d*e*x**2*acosh(c + d*x) - 3*a*b**3*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2 - 3*a*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/2 - a*b**3*e*aco
sh(c + d*x)**3/d - 3*a*b**3*e*acosh(c + d*x)/(2*d) + b**4*c**2*e*acosh(c + d*x)**4/(2*d) + 3*b**4*c**2*e*acosh
(c + d*x)**2/(2*d) + b**4*c*e*x*acosh(c + d*x)**4 + 3*b**4*c*e*x*acosh(c + d*x)**2 + 3*b**4*c*e*x/2 - b**4*c*e
*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**3/d - 3*b**4*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*ac
osh(c + d*x)/(2*d) + b**4*d*e*x**2*acosh(c + d*x)**4/2 + 3*b**4*d*e*x**2*acosh(c + d*x)**2/2 + 3*b**4*d*e*x**2
/4 - b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**3 - 3*b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 - 1)*acosh(c + d*x)/2 - b**4*e*acosh(c + d*x)**4/(4*d) - 3*b**4*e*acosh(c + d*x)**2/(4*d), Ne(d, 0)), (c*e
*x*(a + b*acosh(c))**4, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\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)*(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)