3.26 \(\int \cosh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=68 \[ -\frac{6 \sqrt{a x-1} \sqrt{a x+1}}{a}+x \cosh ^{-1}(a x)^3-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a}+6 x \cosh ^{-1}(a x) \]

[Out]

(-6*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + 6*x*ArcCosh[a*x] - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + x
*ArcCosh[a*x]^3

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Rubi [A]  time = 0.183306, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5654, 5718, 74} \[ -\frac{6 \sqrt{a x-1} \sqrt{a x+1}}{a}+x \cosh ^{-1}(a x)^3-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a}+6 x \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3,x]

[Out]

(-6*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + 6*x*ArcCosh[a*x] - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + x
*ArcCosh[a*x]^3

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 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 \cosh ^{-1}(a x)^3 \, dx &=x \cosh ^{-1}(a x)^3-(3 a) \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3+6 \int \cosh ^{-1}(a x) \, dx\\ &=6 x \cosh ^{-1}(a x)-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3-(6 a) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{6 \sqrt{-1+a x} \sqrt{1+a x}}{a}+6 x \cosh ^{-1}(a x)-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+x \cosh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0254505, size = 68, normalized size = 1. \[ -\frac{6 \sqrt{a x-1} \sqrt{a x+1}}{a}+x \cosh ^{-1}(a x)^3-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a}+6 x \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]^3,x]

[Out]

(-6*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + 6*x*ArcCosh[a*x] - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + x
*ArcCosh[a*x]^3

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Maple [A]  time = 0.031, size = 61, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax-3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}+6\,ax{\rm arccosh} \left (ax\right )-6\,\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3,x)

[Out]

1/a*(arccosh(a*x)^3*a*x-3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*a*x*arccosh(a*x)-6*(a*x-1)^(1/2)*(a*x+1
)^(1/2))

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Maxima [A]  time = 1.20984, size = 77, normalized size = 1.13 \begin{align*} x \operatorname{arcosh}\left (a x\right )^{3} - \frac{3 \, \sqrt{a^{2} x^{2} - 1} \operatorname{arcosh}\left (a x\right )^{2}}{a} + \frac{6 \,{\left (a x \operatorname{arcosh}\left (a x\right ) - \sqrt{a^{2} x^{2} - 1}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3,x, algorithm="maxima")

[Out]

x*arccosh(a*x)^3 - 3*sqrt(a^2*x^2 - 1)*arccosh(a*x)^2/a + 6*(a*x*arccosh(a*x) - sqrt(a^2*x^2 - 1))/a

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Fricas [A]  time = 2.47585, size = 205, normalized size = 3.01 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + 6 \, a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 3 \, \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 6 \, \sqrt{a^{2} x^{2} - 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3,x, algorithm="fricas")

[Out]

(a*x*log(a*x + sqrt(a^2*x^2 - 1))^3 + 6*a*x*log(a*x + sqrt(a^2*x^2 - 1)) - 3*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(
a^2*x^2 - 1))^2 - 6*sqrt(a^2*x^2 - 1))/a

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Sympy [A]  time = 0.570065, size = 63, normalized size = 0.93 \begin{align*} \begin{cases} x \operatorname{acosh}^{3}{\left (a x \right )} + 6 x \operatorname{acosh}{\left (a x \right )} - \frac{3 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{a} - \frac{6 \sqrt{a^{2} x^{2} - 1}}{a} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} x}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3,x)

[Out]

Piecewise((x*acosh(a*x)**3 + 6*x*acosh(a*x) - 3*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/a - 6*sqrt(a**2*x**2 - 1)/a,
Ne(a, 0)), (-I*pi**3*x/8, True))

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Giac [A]  time = 1.36205, size = 132, normalized size = 1.94 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 3 \, a{\left (\frac{\sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{a^{2}} - \frac{2 \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{\sqrt{a^{2} x^{2} - 1}}{a}\right )}}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3,x, algorithm="giac")

[Out]

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