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((c*x-1)^(1/2)*(c*x+1)^(1/2))

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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]

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*}

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Mathematica [A]  time = 0.08, 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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fricas [B]  time = 0.55, size = 74, normalized size = 2.00 \[ \frac {2 \, b c x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + b x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b x - b\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - a}{x} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.01, size = 59, normalized size = 1.59 \[ -\frac {a}{x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{x}-\frac {c b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}} \]

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 = 0.84, size = 30, normalized size = 0.81 \[ -{\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b - \frac {a}{x} \]

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/(c*abs(x))) + arccosh(c*x)/x)*b - a/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))/x^2,x)

[Out]

int((a + b*acosh(c*x))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, dx \]

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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