Integrand size = 21, antiderivative size = 34 \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=-2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]
-2*arcsinh(a*x)*arctanh(a*x+(a^2*x^2+1)^(1/2))-polylog(2,-a*x-(a^2*x^2+1)^ (1/2))+polylog(2,a*x+(a^2*x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\text {arcsinh}(a x) \left (\log \left (1-e^{-\text {arcsinh}(a x)}\right )-\log \left (1+e^{-\text {arcsinh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right ) \]
ArcSinh[a*x]*(Log[1 - E^(-ArcSinh[a*x])] - Log[1 + E^(-ArcSinh[a*x])]) + P olyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6231, 3042, 26, 4670, 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.
\(\displaystyle \int \frac {\text {arcsinh}(a x)}{x \sqrt {a^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 6231 |
\(\displaystyle \int \frac {\text {arcsinh}(a x)}{a x}d\text {arcsinh}(a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \text {arcsinh}(a x) \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \text {arcsinh}(a x) \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle i \left (i \int \log \left (1-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-i \int \log \left (1+e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle i \left (i \int e^{-\text {arcsinh}(a x)} \log \left (1-e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}-i \int e^{-\text {arcsinh}(a x)} \log \left (1+e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}+2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle i \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )\) |
I*((2*I)*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + I*PolyLog[2, -E^ArcSinh[a* x]] - I*PolyLog[2, E^ArcSinh[a*x]])
3.2.82.3.1 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[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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.62
method | result | size |
default | \(\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )\) | \(89\) |
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))
\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x \sqrt {a^{2} x^{2} + 1}}\, dx \]
\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]
Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x\,\sqrt {a^2\,x^2+1}} \,d x \]