Integrand size = 21, antiderivative size = 80 \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=-\frac {a}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 x^2}+a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]
-1/2*a/x+a^2*arcsinh(a*x)*arctanh(a*x+(a^2*x^2+1)^(1/2))+1/2*a^2*polylog(2 ,-a*x-(a^2*x^2+1)^(1/2))-1/2*a^2*polylog(2,a*x+(a^2*x^2+1)^(1/2))-1/2*arcs inh(a*x)*(a^2*x^2+1)^(1/2)/x^2
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\frac {1}{8} a^2 \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-\text {arcsinh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )+4 \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right ) \]
(a^2*(-2*Coth[ArcSinh[a*x]/2] - ArcSinh[a*x]*Csch[ArcSinh[a*x]/2]^2 - 4*Ar cSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] + 4*ArcSinh[a*x]*Log[1 + E^(-ArcSinh [a*x])] - 4*PolyLog[2, -E^(-ArcSinh[a*x])] + 4*PolyLog[2, E^(-ArcSinh[a*x] )] - ArcSinh[a*x]*Sech[ArcSinh[a*x]/2]^2 + 2*Tanh[ArcSinh[a*x]/2]))/8
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6224, 15, 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.
\(\displaystyle \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {a^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 6224 |
\(\displaystyle -\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)}{x \sqrt {a^2 x^2+1}}dx+\frac {1}{2} a \int \frac {1}{x^2}dx-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)}{x \sqrt {a^2 x^2+1}}dx-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
\(\Big \downarrow \) 6231 |
\(\displaystyle -\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)}{a x}d\text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2} a^2 \int i \text {arcsinh}(a x) \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{2} i a^2 \int \text {arcsinh}(a x) \csc (i \text {arcsinh}(a x))d\text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle -\frac {1}{2} i a^2 \left (i \int \log \left (1-e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-i \int \log \left (1+e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)+2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {1}{2} i a^2 \left (i \int e^{-\text {arcsinh}(a x)} \log \left (1-e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}-i \int e^{-\text {arcsinh}(a x)} \log \left (1+e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}+2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {1}{2} i a^2 \left (2 i \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 x^2}-\frac {a}{2 x}\) |
-1/2*a/x - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*x^2) - (I/2)*a^2*((2*I)*Arc Sinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + I*PolyLog[2, -E^ArcSinh[a*x]] - I*Poly Log[2, E^ArcSinh[a*x]])
3.2.84.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
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 = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.88
method | result | size |
default | \(-\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )+a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )}{2 \sqrt {a^{2} x^{2}+1}\, x^{2}}-\frac {a^{2} \operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )}{2}-\frac {a^{2} \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )}{2}+\frac {a^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )}{2}\) | \(150\) |
-1/2/(a^2*x^2+1)^(1/2)*(a^2*x^2*arcsinh(a*x)+a*x*(a^2*x^2+1)^(1/2)+arcsinh (a*x))/x^2-1/2*a^2*arcsinh(a*x)*ln(1-a*x-(a^2*x^2+1)^(1/2))-1/2*a^2*polylo g(2,a*x+(a^2*x^2+1)^(1/2))+1/2*a^2*arcsinh(a*x)*ln(1+a*x+(a^2*x^2+1)^(1/2) )+1/2*a^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))
\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{3} \sqrt {a^{2} x^{2} + 1}}\, dx \]
\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
\[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x^3\,\sqrt {a^2\,x^2+1}} \,d x \]