3.1.4 \(\int x \cosh ^{-1}(a x) \, dx\) [4]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 49, 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 = {5883, 92, 54} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 54

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]

Rule 92

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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*}

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Mathematica [A]
time = 0.02, size = 61, normalized size = 1.24 \begin {gather*} -\frac {a x \sqrt {-1+a x} \sqrt {1+a x}-2 a^2 x^2 \cosh ^{-1}(a x)+2 \tanh ^{-1}\left (\sqrt {\frac {-1+a x}{1+a x}}\right )}{4 a^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(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)]])/a^2

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Maple [A]
time = 1.46, size = 76, normalized size = 1.55

method	result	size
derivativedivides	\(\frac {\frac {a^{2} x^{2} \mathrm {arccosh}\left (a x \right )}{2}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a x \sqrt {a^{2} x^{2}-1}+\ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{4 \sqrt {a^{2} x^{2}-1}}}{a^{2}}\)	\(76\)
default	\(\frac {\frac {a^{2} x^{2} \mathrm {arccosh}\left (a x \right )}{2}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a x \sqrt {a^{2} x^{2}-1}+\ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{4 \sqrt {a^{2} x^{2}-1}}}{a^{2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 56, normalized size = 1.14 \begin {gather*} \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} a\right )}{a^{3}}\right )} \end {gather*}

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)*a)/a^3)

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Fricas [A]
time = 0.35, size = 48, normalized size = 0.98 \begin {gather*} -\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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.09, size = 44, normalized size = 0.90 \begin {gather*} \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 {gather*}

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 = 0.40, size = 70, normalized size = 1.43 \begin {gather*} \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 {gather*}

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)))

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Mupad [B]
time = 0.04, size = 39, normalized size = 0.80 \begin {gather*} x\,\mathrm {acosh}\left (a\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,a^2\,x}\right )-\frac {x\,\sqrt {a\,x-1}\,\sqrt {a\,x+1}}{4\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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