Integrand size = 10, antiderivative size = 52 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \]
1/2*arcsinh(a/x)^2-arcsinh(a/x)*ln(1-(a/x+(a^2/x^2+1)^(1/2))^2)-1/2*polylo g(2,(a/x+(a^2/x^2+1)^(1/2))^2)
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right )^2-\text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right ) \]
ArcSinh[a/x]^2/2 - ArcSinh[a/x]*Log[1 - E^(2*ArcSinh[a/x])] - PolyLog[2, E ^(2*ArcSinh[a/x])]/2
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6284, 3042, 26, 4199, 25, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx\) |
\(\Big \downarrow \) 6284 |
\(\displaystyle -\int \frac {\sqrt {\frac {a^2}{x^2}+1} x \text {arcsinh}\left (\frac {a}{x}\right )}{a}d\text {arcsinh}\left (\frac {a}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -i \text {arcsinh}\left (\frac {a}{x}\right ) \tan \left (i \text {arcsinh}\left (\frac {a}{x}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {a}{x}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \text {arcsinh}\left (\frac {a}{x}\right ) \tan \left (i \text {arcsinh}\left (\frac {a}{x}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {a}{x}\right )\) |
\(\Big \downarrow \) 4199 |
\(\displaystyle i \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (\frac {a}{x}\right )} \text {arcsinh}\left (\frac {a}{x}\right )}{1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}}d\text {arcsinh}\left (\frac {a}{x}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {a}{x}\right )^2\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (\frac {a}{x}\right )} \text {arcsinh}\left (\frac {a}{x}\right )}{1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}}d\text {arcsinh}\left (\frac {a}{x}\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {a}{x}\right )^2\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )d\text {arcsinh}\left (\frac {a}{x}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {a}{x}\right )^2\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle i \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arcsinh}\left (\frac {a}{x}\right )} \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )de^{2 \text {arcsinh}\left (\frac {a}{x}\right )}-\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {a}{x}\right )^2\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )-\frac {1}{2} \text {arcsinh}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {a}{x}\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (\frac {a}{x}\right )^2\right )\) |
I*((-1/2*I)*ArcSinh[a/x]^2 - (2*I)*(-1/2*(ArcSinh[a/x]*Log[1 - E^(2*ArcSin h[a/x])]) - PolyLog[2, E^(2*ArcSinh[a/x])]/4))
3.4.3.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[ArcSinh[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Simp[1/p Subst[Int[ x^n*Coth[x], x], x, ArcSinh[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.19
method | result | size |
derivativedivides | \(\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )^{2}}{2}-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1+\frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, -\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1-\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, \frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )\) | \(114\) |
default | \(\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )^{2}}{2}-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1+\frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, -\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {arcsinh}\left (\frac {a}{x}\right ) \ln \left (1-\frac {a}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}\right )-\operatorname {polylog}\left (2, \frac {a}{x}+\sqrt {\frac {a^{2}}{x^{2}}+1}\right )\) | \(114\) |
1/2*arcsinh(a/x)^2-arcsinh(a/x)*ln(1+a/x+(a^2/x^2+1)^(1/2))-polylog(2,-a/x -(a^2/x^2+1)^(1/2))-arcsinh(a/x)*ln(1-a/x-(a^2/x^2+1)^(1/2))-polylog(2,a/x +(a^2/x^2+1)^(1/2))
\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x} \,d x } \]
\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x}\, dx \]
\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x} \,d x } \]
a*integrate(x*log(x)/(a^3 + a*x^2 + (a^2 + x^2)^(3/2)), x) + log(a + sqrt( a^2 + x^2))*log(x) - 1/2*log(x)^2 - 1/2*log(x)*log(x^2/a^2 + 1) - 1/4*dilo g(-x^2/a^2)
\[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {a}{x}\right )}{x} \,d x \]