3.138 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^2} \, dx\)

Optimal. Leaf size=37 \[ b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )-\frac{a+b \cosh ^{-1}(c x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0239983, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5662, 92, 205} \[ b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )-\frac{a+b \cosh ^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/x^2,x]

[Out]

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

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 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0716771, size = 65, normalized size = 1.76 \[ -\frac{a}{x}+\frac{b c \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b \cosh ^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/x^2,x]

[Out]

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

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Maple [A]  time = 0.004, size = 59, normalized size = 1.6 \begin{align*} -{\frac{a}{x}}-{\frac{b{\rm arccosh} \left (cx\right )}{x}}-{bc\sqrt{cx-1}\sqrt{cx+1}\arctan \left ({\frac{1}{\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((a+b*arccosh(c*x))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.6757, size = 43, normalized size = 1.16 \begin{align*} -{\left (c \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arcosh}\left (c x\right )}{x}\right )} b - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.5959, size = 171, normalized size = 4.62 \begin{align*} \frac{2 \, b c x \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + b x \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (b x - b\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^2,x, algorithm="fricas")

[Out]

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

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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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x**2,x)

[Out]

Integral((a + b*acosh(c*x))/x**2, 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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^2,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/x^2, x)