3.1.16 \(\int \cosh ^{-1}(a x)^2 \, dx\) [16]

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

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 5915, 8} \begin {gather*} x \cosh ^{-1}(a x)^2-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{a}+2 x \end {gather*}

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 8

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

Rule 5879

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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

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*}

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Mathematica [A]
time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} 2 x-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{a}+x \cosh ^{-1}(a x)^2 \end {gather*}

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 = 1.77, size = 39, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {\mathrm {arccosh}\left (a x \right )^{2} a x -2 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+2 a x}{a}\)	\(39\)
default	\(\frac {\mathrm {arccosh}\left (a x \right )^{2} a x -2 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+2 a x}{a}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[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 = 0.26, size = 32, normalized size = 0.82 \begin {gather*} 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 {gather*}

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 = 0.34, size = 59, normalized size = 1.51 \begin {gather*} \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 {gather*}

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 = 0.08, size = 39, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.40, size = 62, normalized size = 1.59 \begin {gather*} 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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(a*x)^2, x)

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