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

Optimal. Leaf size=39 \[ x \cosh ^{-1}(a x)^2-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}+2 x \]

[Out]

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

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Rubi [A]  time = 0.127432, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5654, 5718, 8} \[ x \cosh ^{-1}(a x)^2-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}+2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0198588, size = 39, normalized size = 1. \[ x \cosh ^{-1}(a x)^2-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}+2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 39, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax-2\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}+2\,ax \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13679, size = 43, normalized size = 1.1 \begin{align*} x \operatorname{arcosh}\left (a x\right )^{2} + 2 \, x - \frac{2 \, \sqrt{a^{2} x^{2} - 1} \operatorname{arcosh}\left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.32478, size = 134, normalized size = 3.44 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 2 \, a x - 2 \, \sqrt{a^{2} x^{2} - 1} \log \left (a x + \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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.24437, size = 39, normalized size = 1. \begin{align*} \begin{cases} x \operatorname{acosh}^{2}{\left (a x \right )} + 2 x - \frac{2 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} x}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*acosh(a*x)**2 + 2*x - 2*sqrt(a**2*x**2 - 1)*acosh(a*x)/a, Ne(a, 0)), (-pi**2*x/4, True))

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Giac [A]  time = 1.35347, size = 84, normalized size = 2.15 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 2 \, a{\left (\frac{x}{a} - \frac{\sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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