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]

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

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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^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*}

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Mathematica [A]  time = 0.02, 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]

-1/2*a/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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fricas [A]  time = 0.75, size = 48, normalized size = 1.12 \[ \frac {\sqrt {c^{2} x^{2} - 1} b c x + a x^{2} - b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - a}{2 \, x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{x^{3}}\,{d x} \]

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)

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maple [A]  time = 0.00, size = 52, normalized size = 1.21 \[ c^{2} \left (-\frac {a}{2 c^{2} x^{2}}+b \left (-\frac {\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}}{2 c x}\right )\right ) \]

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 = 0.53, size = 36, normalized size = 0.84 \[ \frac {1}{2} \, b {\left (\frac {\sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\operatorname {arcosh}\left (c x\right )}{x^{2}}\right )} - \frac {a}{2 \, x^{2}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))/x^3,x)

[Out]

int((a + b*acosh(c*x))/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\, dx \]

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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