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

Optimal. Leaf size=215 \[ -\frac{d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac{4 e^2 x}{9 c^2}-\frac{4 e^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{9 c^3}-\frac{d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac{2 d^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{d e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}+\frac{\cosh ^{-1}(c x)^2 (d+e x)^3}{3 e}-\frac{2 e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{9 c}+2 d^2 x+\frac{1}{2} d e x^2+\frac{2 e^2 x^3}{27} \]

[Out]

2*d^2*x + (4*e^2*x)/(9*c^2) + (d*e*x^2)/2 + (2*e^2*x^3)/27 - (2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])
/c - (4*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(9*c^3) - (d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c
*x])/c - (2*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(9*c) - (d^3*ArcCosh[c*x]^2)/(3*e) - (d*e*ArcCo
sh[c*x]^2)/(2*c^2) + ((d + e*x)^3*ArcCosh[c*x]^2)/(3*e)

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Rubi [A]  time = 0.996076, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac{4 e^2 x}{9 c^2}-\frac{4 e^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{9 c^3}-\frac{d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac{2 d^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}-\frac{d e x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c}+\frac{\cosh ^{-1}(c x)^2 (d+e x)^3}{3 e}-\frac{2 e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{9 c}+2 d^2 x+\frac{1}{2} d e x^2+\frac{2 e^2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*ArcCosh[c*x]^2,x]

[Out]

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

Mathematica [A]  time = 0.186733, size = 131, normalized size = 0.61 \[ \frac{c x \left (c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+24 e^2\right )+9 \cosh ^{-1}(c x)^2 \left (2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-3 c d e\right )-6 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x) \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )}{54 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*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))*ArcCosh[c*x] + 9*(-3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcCosh[c*x]^2)/(54*c^3
)

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Maple [A]  time = 0.06, size = 207, normalized size = 1. \begin{align*}{\frac{1}{54\,{c}^{3}} \left ( 18\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{3}{x}^{3}{e}^{2}+54\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{3}{x}^{2}de+54\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{3}x{d}^{2}-12\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{c}^{2}{x}^{2}{e}^{2}-54\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{c}^{2}xde-108\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{c}^{2}{d}^{2}-27\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}cde-24\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{e}^{2}+4\,{e}^{2}{c}^{3}{x}^{3}+27\,{c}^{3}{x}^{2}de+108\,x{c}^{3}{d}^{2}+24\,cx{e}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54/c^3*(18*arccosh(c*x)^2*c^3*x^3*e^2+54*arccosh(c*x)^2*c^3*x^2*d*e+54*arccosh(c*x)^2*c^3*x*d^2-12*arccosh(c
*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2*e^2-54*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x*d*e-108*arccosh(
c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*d^2-27*arccosh(c*x)^2*c*d*e-24*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e
^2+4*e^2*c^3*x^3+27*c^3*x^2*d*e+108*x*c^3*d^2+24*c*x*e^2)

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

[Out]

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

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Fricas [A]  time = 1.98544, size = 358, normalized size = 1.67 \begin{align*} \frac{4 \, c^{3} e^{2} x^{3} + 27 \, c^{3} d e x^{2} + 9 \,{\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt{c^{2} x^{2} - 1} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 12 \,{\left (9 \, c^{3} d^{2} + 2 \, c e^{2}\right )} x}{54 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(4*c^3*e^2*x^3 + 27*c^3*d*e*x^2 + 9*(2*c^3*e^2*x^3 + 6*c^3*d*e*x^2 + 6*c^3*d^2*x - 3*c*d*e)*log(c*x + sqr
t(c^2*x^2 - 1))^2 - 6*(2*c^2*e^2*x^2 + 9*c^2*d*e*x + 18*c^2*d^2 + 4*e^2)*sqrt(c^2*x^2 - 1)*log(c*x + sqrt(c^2*
x^2 - 1)) + 12*(9*c^3*d^2 + 2*c*e^2)*x)/c^3

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Sympy [A]  time = 1.49982, size = 223, normalized size = 1.04 \begin{align*} \begin{cases} d^{2} x \operatorname{acosh}^{2}{\left (c x \right )} + 2 d^{2} x + d e x^{2} \operatorname{acosh}^{2}{\left (c x \right )} + \frac{d e x^{2}}{2} + \frac{e^{2} x^{3} \operatorname{acosh}^{2}{\left (c x \right )}}{3} + \frac{2 e^{2} x^{3}}{27} - \frac{2 d^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{d e x \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{2 e^{2} x^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{9 c} - \frac{d e \operatorname{acosh}^{2}{\left (c x \right )}}{2 c^{2}} + \frac{4 e^{2} x}{9 c^{2}} - \frac{4 e^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{9 c^{3}} & \text{for}\: c \neq 0 \\- \frac{\pi ^{2} \left (d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3}\right )}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2} \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)^2*arccosh(c*x)^2,x, algorithm="giac")

[Out]

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