Integrand size = 8, antiderivative size = 77 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {a \sqrt {1+a^2 x^2}}{20 x^4}+\frac {3 a^3 \sqrt {1+a^2 x^2}}{40 x^2}-\frac {\text {arcsinh}(a x)}{5 x^5}-\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \] Output:
-1/20*a*(a^2*x^2+1)^(1/2)/x^4+3/40*a^3*(a^2*x^2+1)^(1/2)/x^2-1/5*arcsinh(a *x)/x^5-3/40*a^5*arctanh((a^2*x^2+1)^(1/2))
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 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {\text {arcsinh}(a x)}{5 x^5}-\frac {1}{5} a^5 \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1+a^2 x^2\right ) \] Input:
Integrate[ArcSinh[a*x]/x^6,x]
Output:
-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
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6191, 243, 52, 52, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\text {arcsinh}(a x)}{x^6} \, dx\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {1}{5} a \int \frac {1}{x^5 \sqrt {a^2 x^2+1}}dx-\frac {\text {arcsinh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{10} a \int \frac {1}{x^6 \sqrt {a^2 x^2+1}}dx^2-\frac {\text {arcsinh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \int \frac {1}{x^4 \sqrt {a^2 x^2+1}}dx^2-\frac {\sqrt {a^2 x^2+1}}{2 x^4}\right )-\frac {\text {arcsinh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \left (-\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {a^2 x^2+1}}dx^2-\frac {\sqrt {a^2 x^2+1}}{x^2}\right )-\frac {\sqrt {a^2 x^2+1}}{2 x^4}\right )-\frac {\text {arcsinh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \left (-\int \frac {1}{\frac {x^4}{a^2}-\frac {1}{a^2}}d\sqrt {a^2 x^2+1}-\frac {\sqrt {a^2 x^2+1}}{x^2}\right )-\frac {\sqrt {a^2 x^2+1}}{2 x^4}\right )-\frac {\text {arcsinh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \left (a^2 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {\sqrt {a^2 x^2+1}}{x^2}\right )-\frac {\sqrt {a^2 x^2+1}}{2 x^4}\right )-\frac {\text {arcsinh}(a x)}{5 x^5}\) |
Input:
Int[ArcSinh[a*x]/x^6,x]
Output:
-1/5*ArcSinh[a*x]/x^5 + (a*(-1/2*Sqrt[1 + a^2*x^2]/x^4 - (3*a^2*(-(Sqrt[1 + a^2*x^2]/x^2) + a^2*ArcTanh[Sqrt[1 + a^2*x^2]]))/4))/10
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[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]
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(a^{5} \left (-\frac {\operatorname {arcsinh}\left (x a \right )}{5 x^{5} a^{5}}-\frac {\sqrt {a^{2} x^{2}+1}}{20 x^{4} a^{4}}+\frac {3 \sqrt {a^{2} x^{2}+1}}{40 x^{2} a^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{40}\right )\) | \(70\) |
default | \(a^{5} \left (-\frac {\operatorname {arcsinh}\left (x a \right )}{5 x^{5} a^{5}}-\frac {\sqrt {a^{2} x^{2}+1}}{20 x^{4} a^{4}}+\frac {3 \sqrt {a^{2} x^{2}+1}}{40 x^{2} a^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{40}\right )\) | \(70\) |
parts | \(-\frac {\operatorname {arcsinh}\left (x a \right )}{5 x^{5}}+\frac {a \left (-\frac {\sqrt {a^{2} x^{2}+1}}{4 x^{4}}-\frac {3 a^{2} \left (-\frac {\sqrt {a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}\right )}{4}\right )}{5}\) | \(70\) |
Input:
int(arcsinh(x*a)/x^6,x,method=_RETURNVERBOSE)
Output:
a^5*(-1/5/x^5/a^5*arcsinh(x*a)-1/20/x^4/a^4*(a^2*x^2+1)^(1/2)+3/40/x^2/a^2 *(a^2*x^2+1)^(1/2)-3/40*arctanh(1/(a^2*x^2+1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (63) = 126\).
Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.68 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {3 \, a^{5} x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{5} x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 8 \, x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - 8 \, {\left (x^{5} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (3 \, a^{3} x^{3} - 2 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{40 \, x^{5}} \] Input:
integrate(arcsinh(a*x)/x^6,x, algorithm="fricas")
Output:
-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
\[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{6}}\, dx \] Input:
integrate(asinh(a*x)/x**6,x)
Output:
Integral(asinh(a*x)/x**6, x)
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.82 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {1}{40} \, {\left (3 \, a^{4} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {3 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{x^{4}}\right )} a - \frac {\operatorname {arsinh}\left (a x\right )}{5 \, x^{5}} \] Input:
integrate(arcsinh(a*x)/x^6,x, algorithm="maxima")
Output:
-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
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=-\frac {3 \, a^{6} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{6} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) - \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6} - 5 \, \sqrt {a^{2} x^{2} + 1} a^{6}\right )}}{a^{4} x^{4}}}{80 \, a} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{5 \, x^{5}} \] Input:
integrate(arcsinh(a*x)/x^6,x, algorithm="giac")
Output:
-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
Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^6} \,d x \] Input:
int(asinh(a*x)/x^6,x)
Output:
int(asinh(a*x)/x^6, x)
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.19 \[ \int \frac {\text {arcsinh}(a x)}{x^6} \, dx=\frac {-8 \mathit {asinh} \left (a x \right )+3 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-2 \sqrt {a^{2} x^{2}+1}\, a x +3 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x -1\right ) a^{5} x^{5}-3 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x +1\right ) a^{5} x^{5}}{40 x^{5}} \] Input:
int(asinh(a*x)/x^6,x)
Output:
( - 8*asinh(a*x) + 3*sqrt(a**2*x**2 + 1)*a**3*x**3 - 2*sqrt(a**2*x**2 + 1) *a*x + 3*log(sqrt(a**2*x**2 + 1) + a*x - 1)*a**5*x**5 - 3*log(sqrt(a**2*x* *2 + 1) + a*x + 1)*a**5*x**5)/(40*x**5)