3.3 \(\int x^2 \cosh ^{-1}(a x) \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{9 a^3}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0225147, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5662, 100, 12, 74} \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{9 a^3}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCosh[a*x],x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) - (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a) + (x^3*ArcCosh[a*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 \cosh ^{-1}(a x) \, dx &=\frac{1}{3} x^3 \cosh ^{-1}(a x)-\frac{1}{3} a \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)-\frac{\int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a}\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)-\frac{2 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{9 a}+\frac{1}{3} x^3 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0229695, size = 46, normalized size = 0.71 \[ \frac{1}{3} x^3 \cosh ^{-1}(a x)-\frac{\sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2+2\right )}{9 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCosh[a*x],x]

[Out]

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

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Maple [A]  time = 0.009, size = 43, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3}{\rm arccosh} \left (ax\right )}{3}}-{\frac{{a}^{2}{x}^{2}+2}{9}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccosh(a*x),x)

[Out]

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

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Maxima [A]  time = 1.11093, size = 65, normalized size = 1. \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcosh}\left (a x\right ) - \frac{1}{9} \, a{\left (\frac{\sqrt{a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{a^{2} x^{2} - 1}}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccosh(a*x) - 1/9*a*(sqrt(a^2*x^2 - 1)*x^2/a^2 + 2*sqrt(a^2*x^2 - 1)/a^4)

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Fricas [A]  time = 2.40158, size = 113, normalized size = 1.74 \begin{align*} \frac{3 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (a^{2} x^{2} + 2\right )} \sqrt{a^{2} x^{2} - 1}}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.619472, size = 54, normalized size = 0.83 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acosh}{\left (a x \right )}}{3} - \frac{x^{2} \sqrt{a^{2} x^{2} - 1}}{9 a} - \frac{2 \sqrt{a^{2} x^{2} - 1}}{9 a^{3}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{3}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acosh(a*x),x)

[Out]

Piecewise((x**3*acosh(a*x)/3 - x**2*sqrt(a**2*x**2 - 1)/(9*a) - 2*sqrt(a**2*x**2 - 1)/(9*a**3), Ne(a, 0)), (I*
pi*x**3/6, True))

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Giac [A]  time = 1.24391, size = 70, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{a^{2} x^{2} - 1}}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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