\(\int \frac {1}{\text {arcsinh}(a x)} \, dx\) [48]

Optimal result

Integrand size = 6, antiderivative size = 9 \[ \int \frac {1}{\text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a x))}{a} \]

[Out]

Chi(arcsinh(a*x))/a

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5774, 3382} \[ \int \frac {1}{\text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a x))}{a} \]

[In]

Int[ArcSinh[a*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a*x]]/a

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5774

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a x))}{a} \]

[In]

Integrate[ArcSinh[a*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a*x]]/a

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{a}\)	\(10\)
default	\(\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{a}\)	\(10\)

[In]

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

[Out]

Chi(arcsinh(a*x))/a

Fricas [F]

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

[In]

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

[Out]

integral(1/arcsinh(a*x), x)

Sympy [F]

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

[In]

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

[Out]

Integral(1/asinh(a*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/arcsinh(a*x), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/arcsinh(a*x), x)

Mupad [F(-1)]

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

[In]

int(1/asinh(a*x),x)

[Out]

int(1/asinh(a*x), x)