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

Optimal. Leaf size=175 \[ \frac{3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac{3 b e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{3 b^3 e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}}{8 d}-\frac{3 b^3 e \cosh ^{-1}(c+d x)}{8 d} \]

[Out]

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

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Rubi [A]  time = 0.358665, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5866, 12, 5662, 5759, 5676, 90, 52} \[ \frac{3 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac{3 b e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^3}{4 d}-\frac{3 b^3 e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}}{8 d}-\frac{3 b^3 e \cosh ^{-1}(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^3*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(8*d) - (3*b^3*e*ArcCosh[c + d*x])/(8*d) + (3*b^2*e*
(c + d*x)^2*(a + b*ArcCosh[c + d*x]))/(4*d) - (3*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*Arc
Cosh[c + d*x])^2)/(4*d) - (e*(a + b*ArcCosh[c + d*x])^3)/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^3)/(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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.301525, size = 244, normalized size = 1.39 \[ \frac{e \left (2 a \left (2 a^2+3 b^2\right ) (c+d x)^2-3 b \left (2 a^2+b^2\right ) \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}-3 b \left (2 a^2+b^2\right ) \log \left (\sqrt{c+d x-1} \sqrt{c+d x+1}+c+d x\right )-6 b (c+d x) \cosh ^{-1}(c+d x) \left (-2 a^2 (c+d x)+2 a b \sqrt{c+d x-1} \sqrt{c+d x+1}-b^2 (c+d x)\right )+6 b^2 \cosh ^{-1}(c+d x)^2 \left (2 a (c+d x)^2-a-b \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)\right )+2 b^3 \left (2 (c+d x)^2-1\right ) \cosh ^{-1}(c+d x)^3\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.036, size = 605, normalized size = 3.5 \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))^3,x)

[Out]

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

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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))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.51384, size = 891, normalized size = 5.09 \begin{align*} \frac{2 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} d^{2} e x^{2} + 4 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} c d e x + 2 \,{\left (2 \, b^{3} d^{2} e x^{2} + 4 \, b^{3} c d e x +{\left (2 \, b^{3} c^{2} - b^{3}\right )} e\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + 6 \,{\left (2 \, a b^{2} d^{2} e x^{2} + 4 \, a b^{2} c d e x +{\left (2 \, a b^{2} c^{2} - a b^{2}\right )} e -{\left (b^{3} d e x + 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} + 3 \,{\left (2 \,{\left (2 \, a^{2} b + b^{3}\right )} d^{2} e x^{2} + 4 \,{\left (2 \, a^{2} b + b^{3}\right )} c d e x -{\left (2 \, a^{2} b + b^{3} - 2 \,{\left (2 \, a^{2} b + b^{3}\right )} c^{2}\right )} e - 4 \,{\left (a b^{2} d e x + a b^{2} 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 \,{\left ({\left (2 \, a^{2} b + b^{3}\right )} d e x +{\left (2 \, a^{2} b + b^{3}\right )} c e\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{8 \, 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))^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.93132, size = 685, normalized size = 3.91 \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))**3,x)

[Out]

Piecewise((a**3*c*e*x + a**3*d*e*x**2/2 + 3*a**2*b*c**2*e*acosh(c + d*x)/(2*d) + 3*a**2*b*c*e*x*acosh(c + d*x)
- 3*a**2*b*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(4*d) + 3*a**2*b*d*e*x**2*acosh(c + d*x)/2 - 3*a**2*b*e*x
*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/4 - 3*a**2*b*e*acosh(c + d*x)/(4*d) + 3*a*b**2*c**2*e*acosh(c + d*x)**2/
(2*d) + 3*a*b**2*c*e*x*acosh(c + d*x)**2 + 3*a*b**2*c*e*x/2 - 3*a*b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1
)*acosh(c + d*x)/(2*d) + 3*a*b**2*d*e*x**2*acosh(c + d*x)**2/2 + 3*a*b**2*d*e*x**2/4 - 3*a*b**2*e*x*sqrt(c**2
+ 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/2 - 3*a*b**2*e*acosh(c + d*x)**2/(4*d) + b**3*c**2*e*acosh(c + d*x)*
*3/(2*d) + 3*b**3*c**2*e*acosh(c + d*x)/(4*d) + b**3*c*e*x*acosh(c + d*x)**3 + 3*b**3*c*e*x*acosh(c + d*x)/2 -
3*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/(4*d) - 3*b**3*c*e*sqrt(c**2 + 2*c*d*x + d*
*2*x**2 - 1)/(8*d) + b**3*d*e*x**2*acosh(c + d*x)**3/2 + 3*b**3*d*e*x**2*acosh(c + d*x)/4 - 3*b**3*e*x*sqrt(c*
*2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)**2/4 - 3*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/8 - b**3*e
*acosh(c + d*x)**3/(4*d) - 3*b**3*e*acosh(c + d*x)/(8*d), Ne(d, 0)), (c*e*x*(a + b*acosh(c))**3, 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 )}^{3}\,{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))^3,x, algorithm="giac")

[Out]

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