\(\int \frac {1}{\text {arcsinh}(a+b x)} \, dx\) [83]

Optimal result

Integrand size = 8, antiderivative size = 11 \[ \int \frac {1}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a+b x))}{b} \]

[Out]

Chi(arcsinh(b*x+a))/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5858, 5774, 3382} \[ \int \frac {1}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a+b x))}{b} \]

[In]

Int[ArcSinh[a + b*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a + b*x]]/b

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]

Rule 5858

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Chi}(\text {arcsinh}(a+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Chi}(\text {arcsinh}(a+b x))}{b} \]

[In]

Integrate[ArcSinh[a + b*x]^(-1),x]

[Out]

CoshIntegral[ArcSinh[a + b*x]]/b

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b}\)	\(12\)
default	\(\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b}\)	\(12\)

[In]

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

[Out]

Chi(arcsinh(b*x+a))/b

Fricas [F]

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

[In]

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

[Out]

integral(1/arcsinh(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(1/asinh(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/arcsinh(b*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/arcsinh(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(1/asinh(a + b*x),x)

[Out]

int(1/asinh(a + b*x), x)