\(\int \frac {\text {arcsinh}(a x)}{x^4} \, dx\) [9]

Optimal result

Integrand size = 8, antiderivative size = 54 \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=-\frac {a \sqrt {1+a^2 x^2}}{6 x^2}-\frac {\text {arcsinh}(a x)}{3 x^3}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5776, 272, 44, 65, 214} \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=-\frac {a \sqrt {a^2 x^2+1}}{6 x^2}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {\text {arcsinh}(a x)}{3 x^3} \]

[In]

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

[Out]

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

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 214

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

Rule 272

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 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*} \text {integral}& = -\frac {\text {arcsinh}(a x)}{3 x^3}+\frac {1}{3} a \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\text {arcsinh}(a x)}{3 x^3}+\frac {1}{6} a \text {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 {\text {arcsinh}(a x)}{3 x^3}-\frac {1}{12} a^3 \text {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 {\text {arcsinh}(a x)}{3 x^3}-\frac {1}{6} a \text {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 {\text {arcsinh}(a x)}{3 x^3}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=-\frac {a \sqrt {1+a^2 x^2}}{6 x^2}-\frac {\text {arcsinh}(a x)}{3 x^3}+\frac {1}{6} a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89

method	result	size
parts	\(-\frac {\operatorname {arcsinh}\left (a x \right )}{3 x^{3}}+\frac {a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\right )}{3}\)	\(48\)
derivativedivides	\(a^{3} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {\sqrt {a^{2} x^{2}+1}}{6 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{6}\right )\)	\(51\)
default	\(a^{3} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {\sqrt {a^{2} x^{2}+1}}{6 a^{2} x^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{6}\right )\)	\(51\)

[In]

int(arcsinh(a*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (44) = 88\).

Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.17 \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=\frac {a^{3} x^{3} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - a^{3} x^{3} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 2 \, x^{3} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} a x + 2 \, {\left (x^{3} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{6 \, x^{3}} \]

[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]

\[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

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

[Out]

Integral(asinh(a*x)/x**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=\frac {1}{6} \, {\left (a^{2} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{x^{2}}\right )} a - \frac {\operatorname {arsinh}\left (a x\right )}{3 \, x^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=\frac {a^{4} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - a^{4} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) - \frac {2 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{x^{2}}}{12 \, a} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{3 \, x^{3}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^4} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^4} \,d x \]

[In]

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

[Out]

int(asinh(a*x)/x^4, x)