3.1.8 \(\int \frac {\sinh ^{-1}(a x)}{x^3} \, dx\) [8]

Optimal. Leaf size=33 \[ -\frac {a \sqrt {1+a^2 x^2}}{2 x}-\frac {\sinh ^{-1}(a x)}{2 x^2} \]

[Out]

-1/2*arcsinh(a*x)/x^2-1/2*a*(a^2*x^2+1)^(1/2)/x

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5776, 270} \begin {gather*} -\frac {a \sqrt {a^2 x^2+1}}{2 x}-\frac {\sinh ^{-1}(a x)}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]/x^3,x]

[Out]

-1/2*(a*Sqrt[1 + a^2*x^2])/x - ArcSinh[a*x]/(2*x^2)

Rule 270

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

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.85 \begin {gather*} -\frac {a x \sqrt {1+a^2 x^2}+\sinh ^{-1}(a x)}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]/x^3,x]

[Out]

-1/2*(a*x*Sqrt[1 + a^2*x^2] + ArcSinh[a*x])/x^2

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Maple [A]
time = 0.19, size = 37, normalized size = 1.12

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arcsinh \left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{2 a x}\right )\)	\(37\)
default	\(a^{2} \left (-\frac {\arcsinh \left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{2 a x}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2/a^2/x^2*arcsinh(a*x)-1/2/a/x*(a^2*x^2+1)^(1/2))

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Maxima [A]
time = 0.25, size = 27, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {a^{2} x^{2} + 1} a}{2 \, x} - \frac {\operatorname {arsinh}\left (a x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^3,x, algorithm="maxima")

[Out]

-1/2*sqrt(a^2*x^2 + 1)*a/x - 1/2*arcsinh(a*x)/x^2

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Fricas [A]
time = 0.36, size = 36, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {a^{2} x^{2} + 1} a x + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)/x**3,x)

[Out]

Integral(asinh(a*x)/x**3, x)

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Giac [A]
time = 0.43, size = 50, normalized size = 1.52 \begin {gather*} \frac {a {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {asinh}\left (a\,x\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)/x^3,x)

[Out]

int(asinh(a*x)/x^3, x)

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