3.10 \(\int \frac{\sinh ^{-1}(a x)}{x^5} \, dx\)

Optimal. Leaf size=56 \[ \frac{a^3 \sqrt{a^2 x^2+1}}{6 x}-\frac{a \sqrt{a^2 x^2+1}}{12 x^3}-\frac{\sinh ^{-1}(a x)}{4 x^4} \]

[Out]

-(a*Sqrt[1 + a^2*x^2])/(12*x^3) + (a^3*Sqrt[1 + a^2*x^2])/(6*x) - ArcSinh[a*x]/(4*x^4)

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Rubi [A]  time = 0.0204082, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5661, 271, 264} \[ \frac{a^3 \sqrt{a^2 x^2+1}}{6 x}-\frac{a \sqrt{a^2 x^2+1}}{12 x^3}-\frac{\sinh ^{-1}(a x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]/x^5,x]

[Out]

-(a*Sqrt[1 + a^2*x^2])/(12*x^3) + (a^3*Sqrt[1 + a^2*x^2])/(6*x) - ArcSinh[a*x]/(4*x^4)

Rule 5661

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]

Rule 271

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

Rule 264

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]

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x^5} \, dx &=-\frac{\sinh ^{-1}(a x)}{4 x^4}+\frac{1}{4} a \int \frac{1}{x^4 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2}}{12 x^3}-\frac{\sinh ^{-1}(a x)}{4 x^4}-\frac{1}{6} a^3 \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2}}{12 x^3}+\frac{a^3 \sqrt{1+a^2 x^2}}{6 x}-\frac{\sinh ^{-1}(a x)}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0115542, size = 40, normalized size = 0.71 \[ \frac{a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-1\right )-3 \sinh ^{-1}(a x)}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]/x^5,x]

[Out]

(a*x*Sqrt[1 + a^2*x^2]*(-1 + 2*a^2*x^2) - 3*ArcSinh[a*x])/(12*x^4)

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Maple [A]  time = 0.003, size = 56, normalized size = 1. \begin{align*}{a}^{4} \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{4\,{a}^{4}{x}^{4}}}-{\frac{1}{12\,{a}^{3}{x}^{3}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{1}{6\,ax}\sqrt{{a}^{2}{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)/x^5,x)

[Out]

a^4*(-1/4*arcsinh(a*x)/a^4/x^4-1/12/a^3/x^3*(a^2*x^2+1)^(1/2)+1/6/a/x*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.17476, size = 66, normalized size = 1.18 \begin{align*} \frac{1}{12} \,{\left (\frac{2 \, \sqrt{a^{2} x^{2} + 1} a^{2}}{x} - \frac{\sqrt{a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac{\operatorname{arsinh}\left (a x\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^5,x, algorithm="maxima")

[Out]

1/12*(2*sqrt(a^2*x^2 + 1)*a^2/x - sqrt(a^2*x^2 + 1)/x^3)*a - 1/4*arcsinh(a*x)/x^4

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Fricas [A]  time = 1.83009, size = 109, normalized size = 1.95 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} - a x\right )} \sqrt{a^{2} x^{2} + 1} - 3 \, \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^5,x, algorithm="fricas")

[Out]

1/12*((2*a^3*x^3 - a*x)*sqrt(a^2*x^2 + 1) - 3*log(a*x + sqrt(a^2*x^2 + 1)))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)/x**5,x)

[Out]

Integral(asinh(a*x)/x**5, x)

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Giac [A]  time = 1.35278, size = 104, normalized size = 1.86 \begin{align*} \frac{{\left (3 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )} a^{3}{\left | a \right |}}{3 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^5,x, algorithm="giac")

[Out]

1/3*(3*(x*abs(a) - sqrt(a^2*x^2 + 1))^2 - 1)*a^3*abs(a)/((x*abs(a) - sqrt(a^2*x^2 + 1))^2 - 1)^3 - 1/4*log(a*x
+ sqrt(a^2*x^2 + 1))/x^4