3.22 \(\int (d+e x)^2 (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=259 \[ -\frac{d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}-\frac{4 b e^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{2 b d^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b d e x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{2 b e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac{4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3 \]

[Out]

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

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Rubi [A]  time = 1.14838, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac{d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}-\frac{4 b e^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{2 b d^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b d e x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{2 b e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac{4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*d^2*x + (4*b^2*e^2*x)/(9*c^2) + (b^2*d*e*x^2)/2 + (2*b^2*e^2*x^3)/27 - (2*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (b*d*e*x*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (2*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos
h[c*x]))/(9*c) - (d^3*(a + b*ArcCosh[c*x])^2)/(3*e) - (d*e*(a + b*ArcCosh[c*x])^2)/(2*c^2) + ((d + e*x)^3*(a +
b*ArcCosh[c*x])^2)/(3*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)^2 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{(2 b c) \int \frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 e}\\ &=\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{(2 b c) \int \left (\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{3 e}\\ &=\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\left (2 b c d^2\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{\left (2 b c d^3\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 e}-(2 b c d e) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{3} \left (2 b c e^2\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{2 b e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}+\left (2 b^2 d^2\right ) \int 1 \, dx+\left (b^2 d e\right ) \int x \, dx-\frac{(b d e) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c}+\frac{1}{9} \left (2 b^2 e^2\right ) \int x^2 \, dx-\frac{\left (4 b e^2\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c}\\ &=2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{4 b e^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{2 b e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}+\frac{\left (4 b^2 e^2\right ) \int 1 \, dx}{9 c^2}\\ &=2 b^2 d^2 x+\frac{4 b^2 e^2 x}{9 c^2}+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{4 b e^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{b d e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{2 b e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac{d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}\\ \end{align*}

Mathematica [A]  time = 0.611559, size = 360, normalized size = 1.39 \[ a^2 d^2 x+a^2 d e x^2+\frac{1}{3} a^2 e^2 x^3-\frac{b \cosh ^{-1}(c x) \left (b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )-6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{9 c^3}-\frac{a b d e \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{c^2}-\frac{4 a b e^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c^3}-\frac{2 a b d^2 \sqrt{c x-1} \sqrt{c x+1}}{c}-\frac{a b d e x \sqrt{c x-1} \sqrt{c x+1}}{c}-\frac{2 a b e^2 x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c}+\frac{1}{6} b^2 \cosh ^{-1}(c x)^2 \left (2 x \left (3 d^2+3 d e x+e^2 x^2\right )-\frac{3 d e}{c^2}\right )+\frac{4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d^2*x + 2*b^2*d^2*x + (4*b^2*e^2*x)/(9*c^2) + a^2*d*e*x^2 + (b^2*d*e*x^2)/2 + (a^2*e^2*x^3)/3 + (2*b^2*e^2
*x^3)/27 - (2*a*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (4*a*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3) - (a*
b*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (2*a*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) - (b*(-6*a*c^3*x*
(3*d^2 + 3*d*e*x + e^2*x^2) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*Arc
Cosh[c*x])/(9*c^3) + (b^2*((-3*d*e)/c^2 + 2*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcCosh[c*x]^2)/6 - (a*b*d*e*Log[c*
x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/c^2

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Maple [B]  time = 0.052, size = 517, normalized size = 2. \begin{align*} -{\frac{{b}^{2}{\rm arccosh} \left (cx\right )xde}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{abedx}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{2\,ab{d}^{3}}{3\,e}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}+{\frac{{a}^{2}{e}^{2}{x}^{3}}{3}}+{a}^{2}x{d}^{2}+2\,{b}^{2}{d}^{2}x-{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}de}{2\,{c}^{2}}}+{\frac{2\,{b}^{2}{e}^{2}{x}^{3}}{27}}-2\,{\frac{ab\sqrt{cx-1}\sqrt{cx+1}{d}^{2}}{c}}+{\frac{{a}^{2}{d}^{3}}{3\,e}}+{\frac{{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{x}^{3}{e}^{2}}{3}}+{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}x{d}^{2}+{a}^{2}e{x}^{2}d-{\frac{abed}{{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}+{b}^{2} \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{x}^{2}de+2\,ab{\rm arccosh} \left (cx\right )x{d}^{2}+{\frac{2\,ab{\rm arccosh} \left (cx\right ){d}^{3}}{3\,e}}+{\frac{2\,ab{e}^{2}{\rm arccosh} \left (cx\right ){x}^{3}}{3}}-{\frac{2\,ab{e}^{2}{x}^{2}}{9\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{2\,{b}^{2}{\rm arccosh} \left (cx\right ){x}^{2}{e}^{2}}{9\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{4\,{b}^{2}{\rm arccosh} \left (cx\right ){e}^{2}}{9\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-2\,{\frac{{b}^{2}{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{d}^{2}}{c}}-{\frac{4\,ab{e}^{2}}{9\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}+2\,abe{\rm arccosh} \left (cx\right ){x}^{2}d+{\frac{4\,{b}^{2}{e}^{2}x}{9\,{c}^{2}}}+{\frac{{b}^{2}de{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \operatorname{arcosh}\left (c x\right )^{2} + a^{2} d e x^{2} +{\left (2 \, x^{2} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} + \frac{\log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d e + \frac{2}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b e^{2} + 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} - 1} \operatorname{arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} a b d^{2}}{c} + \frac{1}{3} \,{\left (b^{2} e^{2} x^{3} + 3 \, b^{2} d e x^{2}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} - \int \frac{2 \,{\left (b^{2} c^{3} e^{2} x^{5} + 3 \, b^{2} c^{3} d e x^{4} - b^{2} c e^{2} x^{3} - 3 \, b^{2} c d e x^{2} +{\left (b^{2} c^{2} e^{2} x^{4} + 3 \, b^{2} c^{2} d e x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{3 \,{\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)^2*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

1/3*a^2*e^2*x^3 + b^2*d^2*x*arccosh(c*x)^2 + a^2*d*e*x^2 + (2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 +
log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d*e + 2/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2
*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*a*b*e^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2
*d^2*x + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^2/c + 1/3*(b^2*e^2*x^3 + 3*b^2*d*e*x^2)*log(c*x + sqrt
(c*x + 1)*sqrt(c*x - 1))^2 - integrate(2/3*(b^2*c^3*e^2*x^5 + 3*b^2*c^3*d*e*x^4 - b^2*c*e^2*x^3 - 3*b^2*c*d*e*
x^2 + (b^2*c^2*e^2*x^4 + 3*b^2*c^2*d*e*x^3)*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 = 2.33087, size = 687, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \,{\left (2 \, a^{2} + b^{2}\right )} c^{3} d e x^{2} + 9 \,{\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x - 3 \, b^{2} c d e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 6 \,{\left (9 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} + 4 \, b^{2} c e^{2}\right )} x + 6 \,{\left (6 \, a b c^{3} e^{2} x^{3} + 18 \, a b c^{3} d e x^{2} + 18 \, a b c^{3} d^{2} x - 9 \, a b c d e -{\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} + 4 \, b^{2} e^{2}\right )} \sqrt{c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 6 \,{\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} + 4 \, a b e^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{54 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(2*(9*a^2 + 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 + b^2)*c^3*d*e*x^2 + 9*(2*b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2
+ 6*b^2*c^3*d^2*x - 3*b^2*c*d*e)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 6*(9*(a^2 + 2*b^2)*c^3*d^2 + 4*b^2*c*e^2)*x
+ 6*(6*a*b*c^3*e^2*x^3 + 18*a*b*c^3*d*e*x^2 + 18*a*b*c^3*d^2*x - 9*a*b*c*d*e - (2*b^2*c^2*e^2*x^2 + 9*b^2*c^2*
d*e*x + 18*b^2*c^2*d^2 + 4*b^2*e^2)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(2*a*b*c^2*e^2*x^2 + 9
*a*b*c^2*d*e*x + 18*a*b*c^2*d^2 + 4*a*b*e^2)*sqrt(c^2*x^2 - 1))/c^3

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Sympy [A]  time = 2.45324, size = 461, normalized size = 1.78 \begin{align*} \begin{cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac{a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname{acosh}{\left (c x \right )} + 2 a b d e x^{2} \operatorname{acosh}{\left (c x \right )} + \frac{2 a b e^{2} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{2 a b d^{2} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{a b d e x \sqrt{c^{2} x^{2} - 1}}{c} - \frac{2 a b e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{a b d e \operatorname{acosh}{\left (c x \right )}}{c^{2}} - \frac{4 a b e^{2} \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname{acosh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname{acosh}^{2}{\left (c x \right )} + \frac{b^{2} d e x^{2}}{2} + \frac{b^{2} e^{2} x^{3} \operatorname{acosh}^{2}{\left (c x \right )}}{3} + \frac{2 b^{2} e^{2} x^{3}}{27} - \frac{2 b^{2} d^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{b^{2} d e x \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{2 b^{2} e^{2} x^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{9 c} - \frac{b^{2} d e \operatorname{acosh}^{2}{\left (c x \right )}}{2 c^{2}} + \frac{4 b^{2} e^{2} x}{9 c^{2}} - \frac{4 b^{2} e^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{9 c^{3}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right )^{2} \left (d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 + 2*a*b*d**2*x*acosh(c*x) + 2*a*b*d*e*x**2*acosh(c*x
) + 2*a*b*e**2*x**3*acosh(c*x)/3 - 2*a*b*d**2*sqrt(c**2*x**2 - 1)/c - a*b*d*e*x*sqrt(c**2*x**2 - 1)/c - 2*a*b*
e**2*x**2*sqrt(c**2*x**2 - 1)/(9*c) - a*b*d*e*acosh(c*x)/c**2 - 4*a*b*e**2*sqrt(c**2*x**2 - 1)/(9*c**3) + b**2
*d**2*x*acosh(c*x)**2 + 2*b**2*d**2*x + b**2*d*e*x**2*acosh(c*x)**2 + b**2*d*e*x**2/2 + b**2*e**2*x**3*acosh(c
*x)**2/3 + 2*b**2*e**2*x**3/27 - 2*b**2*d**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/c - b**2*d*e*x*sqrt(c**2*x**2 - 1)
*acosh(c*x)/c - 2*b**2*e**2*x**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(9*c) - b**2*d*e*acosh(c*x)**2/(2*c**2) + 4*b*
*2*e**2*x/(9*c**2) - 4*b**2*e**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(9*c**3), Ne(c, 0)), ((a + I*pi*b/2)**2*(d**2*
x + d*e*x**2 + e**2*x**3/3), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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