3.135 \(\int x (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \cosh ^{-1}(c x)}{4 c^2}-\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.0197901, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5662, 90, 52} \[ \frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \cosh ^{-1}(c x)}{4 c^2}-\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCosh[c*x]),x]

[Out]

-(b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) - (b*ArcCosh[c*x])/(4*c^2) + (x^2*(a + b*ArcCosh[c*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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{2} (b c) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c}\\ &=-\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b \cosh ^{-1}(c x)}{4 c^2}+\frac{1}{2} x^2 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.035222, size = 76, normalized size = 1.38 \[ \frac{a x^2}{2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c x-1}}{\sqrt{c x+1}}\right )}{2 c^2}+\frac{1}{2} b x^2 \cosh ^{-1}(c x)-\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCosh[c*x]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 86, normalized size = 1.6 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}{\rm arccosh} \left (cx\right )}{2}}-{\frac{bx}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b}{4\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccosh(c*x)),x)

[Out]

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

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Maxima [A]  time = 1.15047, size = 101, normalized size = 1.84 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} + \frac{\log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.50037, size = 132, normalized size = 2.4 \begin{align*} \frac{2 \, a c^{2} x^{2} - \sqrt{c^{2} x^{2} - 1} b c x +{\left (2 \, b c^{2} x^{2} - b\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.372885, size = 61, normalized size = 1.11 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{b x \sqrt{c^{2} x^{2} - 1}}{4 c} - \frac{b \operatorname{acosh}{\left (c x \right )}}{4 c^{2}} & \text{for}\: c \neq 0 \\\frac{x^{2} \left (a + \frac{i \pi b}{2}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acosh(c*x)),x)

[Out]

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

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Giac [A]  time = 1.43285, size = 108, normalized size = 1.96 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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