\(\int \frac {\text {arcsinh}(a x)}{x^6} \, dx\) [11]

Optimal result

Integrand size = 8, antiderivative size = 77 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {a \sqrt {1+a^2 x^2}}{20 x^4}+\frac {3 a^3 \sqrt {1+a^2 x^2}}{40 x^2}-\frac {\text {arcsinh}(a x)}{5 x^5}-\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]

[Out]

-1/5*arcsinh(a*x)/x^5-3/40*a^5*arctanh((a^2*x^2+1)^(1/2))-1/20*a*(a^2*x^2+1)^(1/2)/x^4+3/40*a^3*(a^2*x^2+1)^(1
/2)/x^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5776, 272, 44, 65, 214} \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {a \sqrt {a^2 x^2+1}}{20 x^4}-\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {3 a^3 \sqrt {a^2 x^2+1}}{40 x^2}-\frac {\text {arcsinh}(a x)}{5 x^5} \]

[In]

Int[ArcSinh[a*x]/x^6,x]

[Out]

-1/20*(a*Sqrt[1 + a^2*x^2])/x^4 + (3*a^3*Sqrt[1 + a^2*x^2])/(40*x^2) - ArcSinh[a*x]/(5*x^5) - (3*a^5*ArcTanh[S
qrt[1 + a^2*x^2]])/40

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)}{5 x^5}+\frac {1}{5} a \int \frac {1}{x^5 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\text {arcsinh}(a x)}{5 x^5}+\frac {1}{10} a \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {a \sqrt {1+a^2 x^2}}{20 x^4}-\frac {\text {arcsinh}(a x)}{5 x^5}-\frac {1}{40} \left (3 a^3\right ) \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}}{20 x^4}+\frac {3 a^3 \sqrt {1+a^2 x^2}}{40 x^2}-\frac {\text {arcsinh}(a x)}{5 x^5}+\frac {1}{80} \left (3 a^5\right ) \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}}{20 x^4}+\frac {3 a^3 \sqrt {1+a^2 x^2}}{40 x^2}-\frac {\text {arcsinh}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^3\right ) \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}}{20 x^4}+\frac {3 a^3 \sqrt {1+a^2 x^2}}{40 x^2}-\frac {\text {arcsinh}(a x)}{5 x^5}-\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {\text {arcsinh}(a x)}{5 x^5}-\frac {1}{5} a^5 \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1+a^2 x^2\right ) \]

[In]

Integrate[ArcSinh[a*x]/x^6,x]

[Out]

-1/5*ArcSinh[a*x]/x^5 - (a^5*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + a^2*x^2])/5

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(a^{5} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{5 a^{5} x^{5}}-\frac {\sqrt {a^{2} x^{2}+1}}{20 a^{4} x^{4}}+\frac {3 \sqrt {a^{2} x^{2}+1}}{40 a^{2} x^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{40}\right )\)	\(70\)
default	\(a^{5} \left (-\frac {\operatorname {arcsinh}\left (a x \right )}{5 a^{5} x^{5}}-\frac {\sqrt {a^{2} x^{2}+1}}{20 a^{4} x^{4}}+\frac {3 \sqrt {a^{2} x^{2}+1}}{40 a^{2} x^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{40}\right )\)	\(70\)
parts	\(-\frac {\operatorname {arcsinh}\left (a x \right )}{5 x^{5}}+\frac {a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{4 x^{4}}-\frac {3 a^{2} \left (-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )}{5}\)	\(70\)

[In]

int(arcsinh(a*x)/x^6,x,method=_RETURNVERBOSE)

[Out]

a^5*(-1/5*arcsinh(a*x)/a^5/x^5-1/20/a^4/x^4*(a^2*x^2+1)^(1/2)+3/40/a^2/x^2*(a^2*x^2+1)^(1/2)-3/40*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. 129 vs. \(2 (63) = 126\).

Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.68 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {3 \, a^{5} x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{5} x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 8 \, x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - 8 \, {\left (x^{5} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (3 \, a^{3} x^{3} - 2 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{40 \, x^{5}} \]

[In]

integrate(arcsinh(a*x)/x^6,x, algorithm="fricas")

[Out]

-1/40*(3*a^5*x^5*log(-a*x + sqrt(a^2*x^2 + 1) + 1) - 3*a^5*x^5*log(-a*x + sqrt(a^2*x^2 + 1) - 1) - 8*x^5*log(-
a*x + sqrt(a^2*x^2 + 1)) - 8*(x^5 - 1)*log(a*x + sqrt(a^2*x^2 + 1)) - (3*a^3*x^3 - 2*a*x)*sqrt(a^2*x^2 + 1))/x
^5

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{6}}\, dx \]

[In]

integrate(asinh(a*x)/x**6,x)

[Out]

Integral(asinh(a*x)/x**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.82 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {1}{40} \, {\left (3 \, a^{4} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {3 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{x^{4}}\right )} a - \frac {\operatorname {arsinh}\left (a x\right )}{5 \, x^{5}} \]

[In]

integrate(arcsinh(a*x)/x^6,x, algorithm="maxima")

[Out]

-1/40*(3*a^4*arcsinh(1/(a*abs(x))) - 3*sqrt(a^2*x^2 + 1)*a^2/x^2 + 2*sqrt(a^2*x^2 + 1)/x^4)*a - 1/5*arcsinh(a*
x)/x^5

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {3 \, a^{6} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{6} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) - \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6} - 5 \, \sqrt {a^{2} x^{2} + 1} a^{6}\right )}}{a^{4} x^{4}}}{80 \, a} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{5 \, x^{5}} \]

[In]

integrate(arcsinh(a*x)/x^6,x, algorithm="giac")

[Out]

-1/80*(3*a^6*log(sqrt(a^2*x^2 + 1) + 1) - 3*a^6*log(sqrt(a^2*x^2 + 1) - 1) - 2*(3*(a^2*x^2 + 1)^(3/2)*a^6 - 5*
sqrt(a^2*x^2 + 1)*a^6)/(a^4*x^4))/a - 1/5*log(a*x + sqrt(a^2*x^2 + 1))/x^5

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^6} \,d x \]

[In]

int(asinh(a*x)/x^6,x)

[Out]

int(asinh(a*x)/x^6, x)