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

Optimal. Leaf size=49 \[ -\frac{\cosh ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{4 a} \]

[Out]

-(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*a) - ArcCosh[a*x]/(4*a^2) + (x^2*ArcCosh[a*x])/2

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Rubi [A]  time = 0.0152043, antiderivative size = 49, 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 = {5662, 90, 52} \[ -\frac{\cosh ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*a) - ArcCosh[a*x]/(4*a^2) + (x^2*ArcCosh[a*x])/2

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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int x \cosh ^{-1}(a x) \, dx &=\frac{1}{2} x^2 \cosh ^{-1}(a x)-\frac{1}{2} a \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{4 a}+\frac{1}{2} x^2 \cosh ^{-1}(a x)-\frac{\int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{4 a}\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{4 a}-\frac{\cosh ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0246326, size = 61, normalized size = 1.24 \[ -\frac{-2 a^2 x^2 \cosh ^{-1}(a x)+a x \sqrt{a x-1} \sqrt{a x+1}+2 \tanh ^{-1}\left (\sqrt{\frac{a x-1}{a x+1}}\right )}{4 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] - 2*a^2*x^2*ArcCosh[a*x] + 2*ArcTanh[Sqrt[(-1 + a*x)/(1 + a*x)]])/(4*a^2)

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Maple [A]  time = 0.011, size = 77, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}{\rm arccosh} \left (ax\right )}{2}}-{\frac{x}{4\,a}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{1}{4\,{a}^{2}}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( ax+\sqrt{{a}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arccosh(a*x)-1/4*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-1/4/a^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)^(1/2)
*ln(a*x+(a^2*x^2-1)^(1/2))

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Maxima [A]  time = 1.13153, size = 88, normalized size = 1.8 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arcosh}\left (a x\right ) - \frac{1}{4} \, a{\left (\frac{\sqrt{a^{2} x^{2} - 1} x}{a^{2}} + \frac{\log \left (2 \, a^{2} x + 2 \, \sqrt{a^{2} x^{2} - 1} \sqrt{a^{2}}\right )}{\sqrt{a^{2}} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.34593, size = 109, normalized size = 2.22 \begin{align*} -\frac{\sqrt{a^{2} x^{2} - 1} a x -{\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(a^2*x^2 - 1)*a*x - (2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a^2

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Sympy [A]  time = 0.265342, size = 44, normalized size = 0.9 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acosh}{\left (a x \right )}}{2} - \frac{x \sqrt{a^{2} x^{2} - 1}}{4 a} - \frac{\operatorname{acosh}{\left (a x \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{2}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2*acosh(a*x)/2 - x*sqrt(a**2*x**2 - 1)/(4*a) - acosh(a*x)/(4*a**2), Ne(a, 0)), (I*pi*x**2/4, Tru
e))

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Giac [A]  time = 1.27737, size = 95, normalized size = 1.94 \begin{align*} \frac{1}{2} \, x^{2} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{1}{4} \, a{\left (\frac{\sqrt{a^{2} x^{2} - 1} x}{a^{2}} - \frac{\log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right )}{a^{2}{\left | a \right |}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(a*x + sqrt(a^2*x^2 - 1)) - 1/4*a*(sqrt(a^2*x^2 - 1)*x/a^2 - log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1))
)/(a^2*abs(a)))