3.9 \(\int \frac{\sinh ^{-1}(a x)}{x^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a \sqrt{a^2 x^2+1}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{3 x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.032432, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5661, 266, 51, 63, 208} \[ -\frac{a \sqrt{a^2 x^2+1}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]/x^4,x]

[Out]

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

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 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.01014, size = 54, normalized size = 1. \[ -\frac{a \sqrt{a^2 x^2+1}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]/x^4,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 51, normalized size = 0.9 \begin{align*}{a}^{3} \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{3\,{a}^{3}{x}^{3}}}-{\frac{1}{6\,{a}^{2}{x}^{2}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{1}{6}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)/x^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.11922, size = 61, normalized size = 1.13 \begin{align*} \frac{1}{6} \,{\left (a^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{\sqrt{a^{2} x^{2} + 1}}{x^{2}}\right )} a - \frac{\operatorname{arsinh}\left (a x\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^4,x, algorithm="maxima")

[Out]

1/6*(a^2*arcsinh(1/(sqrt(a^2)*abs(x))) - sqrt(a^2*x^2 + 1)/x^2)*a - 1/3*arcsinh(a*x)/x^3

________________________________________________________________________________________

Fricas [B]  time = 2.03025, size = 273, normalized size = 5.06 \begin{align*} \frac{a^{3} x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - a^{3} x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) + 2 \, x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right ) - \sqrt{a^{2} x^{2} + 1} a x + 2 \,{\left (x^{3} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^4,x, algorithm="fricas")

[Out]

1/6*(a^3*x^3*log(-a*x + sqrt(a^2*x^2 + 1) + 1) - a^3*x^3*log(-a*x + sqrt(a^2*x^2 + 1) - 1) + 2*x^3*log(-a*x +
sqrt(a^2*x^2 + 1)) - sqrt(a^2*x^2 + 1)*a*x + 2*(x^3 - 1)*log(a*x + sqrt(a^2*x^2 + 1)))/x^3

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)/x**4,x)

[Out]

Integral(asinh(a*x)/x**4, x)

________________________________________________________________________________________

Giac [A]  time = 1.37784, size = 104, normalized size = 1.93 \begin{align*} -\frac{1}{12} \, a^{3}{\left (\frac{2 \, \sqrt{a^{2} x^{2} + 1}}{a^{2} x^{2}} - \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) + \log \left (\sqrt{a^{2} x^{2} + 1} - 1\right )\right )} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^4,x, algorithm="giac")

[Out]

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