\(\int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx\) [182]

Optimal result

Integrand size = 21, antiderivative size = 34 \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=-2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]

[Out]

-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))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5816, 4267, 2317, 2438} \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=-2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right ) \]

[In]

Int[ArcSinh[a*x]/(x*Sqrt[1 + a^2*x^2]),x]

[Out]

-2*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] - PolyLog[2, -E^ArcSinh[a*x]] + PolyLog[2, E^ArcSinh[a*x]]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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]

Rule 5816

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(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] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arcsinh}(a x)) \\ & = -2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-\text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+\text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\text {arcsinh}(a x) \left (\log \left (1-e^{-\text {arcsinh}(a x)}\right )-\log \left (1+e^{-\text {arcsinh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right ) \]

[In]

Integrate[ArcSinh[a*x]/(x*Sqrt[1 + a^2*x^2]),x]

[Out]

ArcSinh[a*x]*(Log[1 - E^(-ArcSinh[a*x])] - Log[1 + E^(-ArcSinh[a*x])]) + PolyLog[2, -E^(-ArcSinh[a*x])] - Poly
Log[2, E^(-ArcSinh[a*x])]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.62

method	result	size
default	\(\operatorname {arcsinh}\left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (a x \right ) \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )\)	\(89\)

[In]

int(arcsinh(a*x)/x/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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))

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]

[In]

integrate(arcsinh(a*x)/x/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*x^2 + 1)*arcsinh(a*x)/(a^2*x^3 + x), x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{x \sqrt {a^{2} x^{2} + 1}}\, dx \]

[In]

integrate(asinh(a*x)/x/(a**2*x**2+1)**(1/2),x)

[Out]

Integral(asinh(a*x)/(x*sqrt(a**2*x**2 + 1)), x)

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]

[In]

integrate(arcsinh(a*x)/x/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)/(sqrt(a^2*x^2 + 1)*x), x)

Giac [F]

\[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]

[In]

integrate(arcsinh(a*x)/x/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)/(sqrt(a^2*x^2 + 1)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{x\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

int(asinh(a*x)/(x*(a^2*x^2 + 1)^(1/2)),x)

[Out]

int(asinh(a*x)/(x*(a^2*x^2 + 1)^(1/2)), x)