3.8 \(\int (d+e x)^3 \cosh ^{-1}(c x)^2 \, dx\)

Optimal. Leaf size=334 \[ -\frac{3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}+\frac{4 d e^2 x}{3 c^2}-\frac{4 d e^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3}+\frac{3 e^3 x^2}{32 c^2}-\frac{3 e^3 \cosh ^{-1}(c x)^2}{32 c^4}-\frac{3 e^3 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{16 c^3}-\frac{d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac{3 d^2 e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{2 c}-\frac{2 d^3 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{2 d e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c}+\frac{\cosh ^{-1}(c x)^2 (d+e x)^4}{4 e}-\frac{e^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{8 c}+\frac{3}{4} d^2 e x^2+2 d^3 x+\frac{2}{9} d e^2 x^3+\frac{e^3 x^4}{32} \]

[Out]

2*d^3*x + (4*d*e^2*x)/(3*c^2) + (3*d^2*e*x^2)/4 + (3*e^3*x^2)/(32*c^2) + (2*d*e^2*x^3)/9 + (e^3*x^4)/32 - (2*d
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (4*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^3)
- (3*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c) - (3*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[
c*x])/(16*c^3) - (2*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c) - (e^3*x^3*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*ArcCosh[c*x])/(8*c) - (d^4*ArcCosh[c*x]^2)/(4*e) - (3*d^2*e*ArcCosh[c*x]^2)/(4*c^2) - (3*e^3*ArcCosh[
c*x]^2)/(32*c^4) + ((d + e*x)^4*ArcCosh[c*x]^2)/(4*e)

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Rubi [A]  time = 1.46354, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac{3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}+\frac{4 d e^2 x}{3 c^2}-\frac{4 d e^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3}+\frac{3 e^3 x^2}{32 c^2}-\frac{3 e^3 \cosh ^{-1}(c x)^2}{32 c^4}-\frac{3 e^3 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{16 c^3}-\frac{d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac{3 d^2 e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{2 c}-\frac{2 d^3 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{2 d e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c}+\frac{\cosh ^{-1}(c x)^2 (d+e x)^4}{4 e}-\frac{e^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{8 c}+\frac{3}{4} d^2 e x^2+2 d^3 x+\frac{2}{9} d e^2 x^3+\frac{e^3 x^4}{32} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*ArcCosh[c*x]^2,x]

[Out]

2*d^3*x + (4*d*e^2*x)/(3*c^2) + (3*d^2*e*x^2)/4 + (3*e^3*x^2)/(32*c^2) + (2*d*e^2*x^3)/9 + (e^3*x^4)/32 - (2*d
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (4*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^3)
- (3*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c) - (3*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[
c*x])/(16*c^3) - (2*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c) - (e^3*x^3*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*ArcCosh[c*x])/(8*c) - (d^4*ArcCosh[c*x]^2)/(4*e) - (3*d^2*e*ArcCosh[c*x]^2)/(4*c^2) - (3*e^3*ArcCosh[
c*x]^2)/(32*c^4) + ((d + e*x)^4*ArcCosh[c*x]^2)/(4*e)

Rule 5802

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)
*(a + b*ArcCosh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*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, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5822

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

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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> 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*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(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, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

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

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 30

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

Rubi steps

\begin{align*} \int (d+e x)^3 \cosh ^{-1}(c x)^2 \, dx &=\frac{(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}-\frac{c \int \frac{(d+e x)^4 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 e}\\ &=\frac{(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}-\frac{c \int \left (\frac{d^4 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 d^3 e x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 d^2 e^2 x^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 d e^3 x^3 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^4 x^4 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{2 e}\\ &=\frac{(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}-\left (2 c d^3\right ) \int \frac{x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{\left (c d^4\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 e}-\left (3 c d^2 e\right ) \int \frac{x^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\left (2 c d e^2\right ) \int \frac{x^3 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{2} \left (c e^3\right ) \int \frac{x^4 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 d^3 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c}-\frac{3 d^2 e x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{2 c}-\frac{2 d e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c}-\frac{e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{8 c}-\frac{d^4 \cosh ^{-1}(c x)^2}{4 e}+\frac{(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}+\left (2 d^3\right ) \int 1 \, dx+\frac{1}{2} \left (3 d^2 e\right ) \int x \, dx-\frac{\left (3 d^2 e\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c}+\frac{1}{3} \left (2 d e^2\right ) \int x^2 \, dx-\frac{\left (4 d e^2\right ) \int \frac{x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c}+\frac{1}{8} e^3 \int x^3 \, dx-\frac{\left (3 e^3\right ) \int \frac{x^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c}\\ &=2 d^3 x+\frac{3}{4} d^2 e x^2+\frac{2}{9} d e^2 x^3+\frac{e^3 x^4}{32}-\frac{2 d^3 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c}-\frac{4 d e^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^3}-\frac{3 d^2 e x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{2 c}-\frac{3 e^3 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{16 c^3}-\frac{2 d e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c}-\frac{e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{8 c}-\frac{d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac{3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}+\frac{(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}+\frac{\left (4 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac{\left (3 e^3\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^3}+\frac{\left (3 e^3\right ) \int x \, dx}{16 c^2}\\ &=2 d^3 x+\frac{4 d e^2 x}{3 c^2}+\frac{3}{4} d^2 e x^2+\frac{3 e^3 x^2}{32 c^2}+\frac{2}{9} d e^2 x^3+\frac{e^3 x^4}{32}-\frac{2 d^3 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{c}-\frac{4 d e^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^3}-\frac{3 d^2 e x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{2 c}-\frac{3 e^3 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{16 c^3}-\frac{2 d e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c}-\frac{e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{8 c}-\frac{d^4 \cosh ^{-1}(c x)^2}{4 e}-\frac{3 d^2 e \cosh ^{-1}(c x)^2}{4 c^2}-\frac{3 e^3 \cosh ^{-1}(c x)^2}{32 c^4}+\frac{(d+e x)^4 \cosh ^{-1}(c x)^2}{4 e}\\ \end{align*}

Mathematica [A]  time = 0.256192, size = 191, normalized size = 0.57 \[ \frac{c^2 x \left (c^2 \left (216 d^2 e x+576 d^3+64 d e^2 x^2+9 e^3 x^3\right )+3 e^2 (128 d+9 e x)\right )-6 c \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x) \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )+9 \cosh ^{-1}(c x)^2 \left (8 c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*ArcCosh[c*x]^2,x]

[Out]

(c^2*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3)) - 6*c*Sqrt[-1 + c*x]*S
qrt[1 + c*x]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3))*ArcCosh[c*x] + 9*(-24
*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcCosh[c*x]^2)/(288*c^4)

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Maple [A]  time = 0.072, size = 329, normalized size = 1. \begin{align*}{\frac{1}{288\,{c}^{4}} \left ( 72\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{4}{x}^{4}{e}^{3}+288\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{4}{x}^{3}d{e}^{2}+432\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{4}{x}^{2}{d}^{2}e+288\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{4}x{d}^{3}-36\,{\rm arccosh} \left (cx\right )\sqrt{cx+1}\sqrt{cx-1}{c}^{3}{x}^{3}{e}^{3}-192\,{\rm arccosh} \left (cx\right )\sqrt{cx+1}\sqrt{cx-1}{c}^{3}{x}^{2}d{e}^{2}-432\,{\rm arccosh} \left (cx\right )\sqrt{cx+1}\sqrt{cx-1}{c}^{3}x{d}^{2}e-576\,{\rm arccosh} \left (cx\right )\sqrt{cx+1}\sqrt{cx-1}{c}^{3}{d}^{3}-216\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{2}{d}^{2}e-54\,{\rm arccosh} \left (cx\right )\sqrt{cx+1}\sqrt{cx-1}cx{e}^{3}-384\,{\rm arccosh} \left (cx\right )\sqrt{cx+1}\sqrt{cx-1}cd{e}^{2}+9\,{c}^{4}{x}^{4}{e}^{3}+64\,{c}^{4}d{e}^{2}{x}^{3}+216\,{c}^{4}{d}^{2}e{x}^{2}+576\,x{c}^{4}{d}^{3}-27\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{e}^{3}+27\,{c}^{2}{x}^{2}{e}^{3}+384\,x{c}^{2}d{e}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*arccosh(c*x)^2,x)

[Out]

1/288/c^4*(72*arccosh(c*x)^2*c^4*x^4*e^3+288*arccosh(c*x)^2*c^4*x^3*d*e^2+432*arccosh(c*x)^2*c^4*x^2*d^2*e+288
*arccosh(c*x)^2*c^4*x*d^3-36*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3*x^3*e^3-192*arccosh(c*x)*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*c^3*x^2*d*e^2-432*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3*x*d^2*e-576*arccosh(c*x)*(c*x+
1)^(1/2)*(c*x-1)^(1/2)*c^3*d^3-216*arccosh(c*x)^2*c^2*d^2*e-54*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c*x*e^
3-384*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c*d*e^2+9*c^4*x^4*e^3+64*c^4*d*e^2*x^3+216*c^4*d^2*e*x^2+576*x*
c^4*d^3-27*arccosh(c*x)^2*e^3+27*c^2*x^2*e^3+384*x*c^2*d*e^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \,{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} - \int \frac{{\left (c^{3} e^{3} x^{6} + 4 \, c^{3} d e^{2} x^{5} - 6 \, c d^{2} e x^{2} - 4 \, c d^{3} x +{\left (6 \, c^{3} d^{2} e - c e^{3}\right )} x^{4} + 4 \,{\left (c^{3} d^{3} - c d e^{2}\right )} x^{3} +{\left (c^{2} e^{3} x^{5} + 4 \, c^{2} d e^{2} x^{4} + 6 \, c^{2} d^{2} e x^{3} + 4 \, c^{2} d^{3} x^{2}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{2 \,{\left (c^{3} x^{3} +{\left (c^{2} x^{2} - 1\right )} \sqrt{c x + 1} \sqrt{c x - 1} - c x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98508, size = 513, normalized size = 1.54 \begin{align*} \frac{9 \, c^{4} e^{3} x^{4} + 64 \, c^{4} d e^{2} x^{3} + 27 \,{\left (8 \, c^{4} d^{2} e + c^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, c^{4} e^{3} x^{4} + 32 \, c^{4} d e^{2} x^{3} + 48 \, c^{4} d^{2} e x^{2} + 32 \, c^{4} d^{3} x - 24 \, c^{2} d^{2} e - 3 \, e^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (6 \, c^{3} e^{3} x^{3} + 32 \, c^{3} d e^{2} x^{2} + 96 \, c^{3} d^{3} + 64 \, c d e^{2} + 9 \,{\left (8 \, c^{3} d^{2} e + c e^{3}\right )} x\right )} \sqrt{c^{2} x^{2} - 1} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 192 \,{\left (3 \, c^{4} d^{3} + 2 \, c^{2} d e^{2}\right )} x}{288 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="fricas")

[Out]

1/288*(9*c^4*e^3*x^4 + 64*c^4*d*e^2*x^3 + 27*(8*c^4*d^2*e + c^2*e^3)*x^2 + 9*(8*c^4*e^3*x^4 + 32*c^4*d*e^2*x^3
+ 48*c^4*d^2*e*x^2 + 32*c^4*d^3*x - 24*c^2*d^2*e - 3*e^3)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(6*c^3*e^3*x^3 +
32*c^3*d*e^2*x^2 + 96*c^3*d^3 + 64*c*d*e^2 + 9*(8*c^3*d^2*e + c*e^3)*x)*sqrt(c^2*x^2 - 1)*log(c*x + sqrt(c^2*
x^2 - 1)) + 192*(3*c^4*d^3 + 2*c^2*d*e^2)*x)/c^4

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Sympy [A]  time = 3.55514, size = 371, normalized size = 1.11 \begin{align*} \begin{cases} d^{3} x \operatorname{acosh}^{2}{\left (c x \right )} + 2 d^{3} x + \frac{3 d^{2} e x^{2} \operatorname{acosh}^{2}{\left (c x \right )}}{2} + \frac{3 d^{2} e x^{2}}{4} + d e^{2} x^{3} \operatorname{acosh}^{2}{\left (c x \right )} + \frac{2 d e^{2} x^{3}}{9} + \frac{e^{3} x^{4} \operatorname{acosh}^{2}{\left (c x \right )}}{4} + \frac{e^{3} x^{4}}{32} - \frac{2 d^{3} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{3 d^{2} e x \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{2 c} - \frac{2 d e^{2} x^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{3 c} - \frac{e^{3} x^{3} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{8 c} - \frac{3 d^{2} e \operatorname{acosh}^{2}{\left (c x \right )}}{4 c^{2}} + \frac{4 d e^{2} x}{3 c^{2}} + \frac{3 e^{3} x^{2}}{32 c^{2}} - \frac{4 d e^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{3 c^{3}} - \frac{3 e^{3} x \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{16 c^{3}} - \frac{3 e^{3} \operatorname{acosh}^{2}{\left (c x \right )}}{32 c^{4}} & \text{for}\: c \neq 0 \\- \frac{\pi ^{2} \left (d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4}\right )}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*acosh(c*x)**2,x)

[Out]

Piecewise((d**3*x*acosh(c*x)**2 + 2*d**3*x + 3*d**2*e*x**2*acosh(c*x)**2/2 + 3*d**2*e*x**2/4 + d*e**2*x**3*aco
sh(c*x)**2 + 2*d*e**2*x**3/9 + e**3*x**4*acosh(c*x)**2/4 + e**3*x**4/32 - 2*d**3*sqrt(c**2*x**2 - 1)*acosh(c*x
)/c - 3*d**2*e*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(2*c) - 2*d*e**2*x**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(3*c) - e
**3*x**3*sqrt(c**2*x**2 - 1)*acosh(c*x)/(8*c) - 3*d**2*e*acosh(c*x)**2/(4*c**2) + 4*d*e**2*x/(3*c**2) + 3*e**3
*x**2/(32*c**2) - 4*d*e**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(3*c**3) - 3*e**3*x*sqrt(c**2*x**2 - 1)*acosh(c*x)/(
16*c**3) - 3*e**3*acosh(c*x)**2/(32*c**4), Ne(c, 0)), (-pi**2*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x
**4/4)/4, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{3} \operatorname{arcosh}\left (c x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="giac")

[Out]

integrate((e*x + d)^3*arccosh(c*x)^2, x)