\(\int \frac {x}{\text {arcsinh}(a x)} \, dx\) [47]

Optimal result

Integrand size = 8, antiderivative size = 14 \[ \int \frac {x}{\text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(2 \text {arcsinh}(a x))}{2 a^2} \]

[Out]

1/2*Shi(2*arcsinh(a*x))/a^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5780, 5556, 12, 3379} \[ \int \frac {x}{\text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(2 \text {arcsinh}(a x))}{2 a^2} \]

[In]

Int[x/ArcSinh[a*x],x]

[Out]

SinhIntegral[2*ArcSinh[a*x]]/(2*a^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arcsinh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^2} \\ & = \frac {\text {Shi}(2 \text {arcsinh}(a x))}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(2 \text {arcsinh}(a x))}{2 a^2} \]

[In]

Integrate[x/ArcSinh[a*x],x]

[Out]

SinhIntegral[2*ArcSinh[a*x]]/(2*a^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2}}\)	\(13\)
default	\(\frac {\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2}}\)	\(13\)

[In]

int(x/arcsinh(a*x),x,method=_RETURNVERBOSE)

[Out]

1/2*Shi(2*arcsinh(a*x))/a^2

Fricas [F]

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

[In]

integrate(x/arcsinh(a*x),x, algorithm="fricas")

[Out]

integral(x/arcsinh(a*x), x)

Sympy [F]

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

[In]

integrate(x/asinh(a*x),x)

[Out]

Integral(x/asinh(a*x), x)

Maxima [F]

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

[In]

integrate(x/arcsinh(a*x),x, algorithm="maxima")

[Out]

integrate(x/arcsinh(a*x), x)

Giac [F]

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

[In]

integrate(x/arcsinh(a*x),x, algorithm="giac")

[Out]

integrate(x/arcsinh(a*x), x)

Mupad [F(-1)]

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

[In]

int(x/asinh(a*x),x)

[Out]

int(x/asinh(a*x), x)