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

Optimal. Leaf size=71 \[ \frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 b \sqrt{c x-1} \sqrt{c x+1}}{9 c^3}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]

[Out]

(-2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) + (x^3*(a + b*ArcCosh
[c*x]))/3

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Rubi [A]  time = 0.0295427, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5662, 100, 12, 74} \[ \frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 b \sqrt{c x-1} \sqrt{c x+1}}{9 c^3}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCosh[c*x]),x]

[Out]

(-2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) + (x^3*(a + b*ArcCosh
[c*x]))/3

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 100

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 74

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

Rubi steps

\begin{align*} \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{3} (b c) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c}\\ &=-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(2 b) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c}\\ &=-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c}+\frac{1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.043911, size = 54, normalized size = 0.76 \[ \frac{1}{9} \left (3 a x^3-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2+2\right )}{c^3}+3 b x^3 \cosh ^{-1}(c x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*ArcCosh[c*x]),x]

[Out]

(3*a*x^3 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c^2*x^2))/c^3 + 3*b*x^3*ArcCosh[c*x])/9

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Maple [A]  time = 0.004, size = 55, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{{c}^{3}{x}^{3}a}{3}}+b \left ({\frac{{c}^{3}{x}^{3}{\rm arccosh} \left (cx\right )}{3}}-{\frac{{c}^{2}{x}^{2}+2}{9}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(1/3*c^3*x^3*a+b*(1/3*c^3*x^3*arccosh(c*x)-1/9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(c^2*x^2+2)))

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Maxima [A]  time = 1.17177, size = 78, normalized size = 1.1 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{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 )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/9*(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))*b

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Fricas [A]  time = 2.42938, size = 140, normalized size = 1.97 \begin{align*} \frac{3 \, b c^{3} x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 3 \, a c^{3} x^{3} -{\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt{c^{2} x^{2} - 1}}{9 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*b*c^3*x^3*log(c*x + sqrt(c^2*x^2 - 1)) + 3*a*c^3*x^3 - (b*c^2*x^2 + 2*b)*sqrt(c^2*x^2 - 1))/c^3

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Sympy [A]  time = 0.67728, size = 71, normalized size = 1. \begin{align*} \begin{cases} \frac{a x^{3}}{3} + \frac{b x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{b x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{2 b \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\frac{x^{3} \left (a + \frac{i \pi b}{2}\right )}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.45294, size = 84, normalized size = 1.18 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/9*(3*x^3*log(c*x + sqrt(c^2*x^2 - 1)) - ((c^2*x^2 - 1)^(3/2) + 3*sqrt(c^2*x^2 - 1))/c^3)*b