3.139 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0200365, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5662, 95} \[ \frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/x^3,x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (a + b*ArcCosh[c*x])/(2*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 95

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

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^3} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{2 x^2}+\frac{1}{2} (b c) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0175018, size = 48, normalized size = 1.12 \[ -\frac{a}{2 x^2}-\frac{b \cosh ^{-1}(c x)}{2 x^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 52, normalized size = 1.2 \begin{align*}{c}^{2} \left ( -{\frac{a}{2\,{c}^{2}{x}^{2}}}+b \left ( -{\frac{{\rm arccosh} \left (cx\right )}{2\,{c}^{2}{x}^{2}}}+{\frac{1}{2\,cx}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x^3,x)

[Out]

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

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Maxima [A]  time = 1.74016, size = 49, normalized size = 1.14 \begin{align*} \frac{1}{2} \, b{\left (\frac{\sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\operatorname{arcosh}\left (c x\right )}{x^{2}}\right )} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.31892, size = 108, normalized size = 2.51 \begin{align*} \frac{\sqrt{c^{2} x^{2} - 1} b c x + a x^{2} - b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acosh}{\left (c x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x**3,x)

[Out]

Integral((a + b*acosh(c*x))/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/x^3, x)