\(\int \text {arcsinh}(a x) \, dx\) [5]

Optimal result

Integrand size = 4, antiderivative size = 25 \[ \int \text {arcsinh}(a x) \, dx=-\frac {\sqrt {1+a^2 x^2}}{a}+x \text {arcsinh}(a x) \]

[Out]

x*arcsinh(a*x)-(a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5772, 267} \[ \int \text {arcsinh}(a x) \, dx=x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a} \]

[In]

Int[ArcSinh[a*x],x]

[Out]

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

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = x \text {arcsinh}(a x)-a \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{a}+x \text {arcsinh}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \text {arcsinh}(a x) \, dx=-\frac {\sqrt {1+a^2 x^2}}{a}+x \text {arcsinh}(a x) \]

[In]

Integrate[ArcSinh[a*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96

method	result	size
parts	\(x \,\operatorname {arcsinh}\left (a x \right )-\frac {\sqrt {a^{2} x^{2}+1}}{a}\)	\(24\)
derivativedivides	\(\frac {a x \,\operatorname {arcsinh}\left (a x \right )-\sqrt {a^{2} x^{2}+1}}{a}\)	\(26\)
default	\(\frac {a x \,\operatorname {arcsinh}\left (a x \right )-\sqrt {a^{2} x^{2}+1}}{a}\)	\(26\)

[In]

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

[Out]

x*arcsinh(a*x)-(a^2*x^2+1)^(1/2)/a

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \text {arcsinh}(a x) \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1}}{a} \]

[In]

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

[Out]

(a*x*log(a*x + sqrt(a^2*x^2 + 1)) - sqrt(a^2*x^2 + 1))/a

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \text {arcsinh}(a x) \, dx=\begin {cases} x \operatorname {asinh}{\left (a x \right )} - \frac {\sqrt {a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(asinh(a*x),x)

[Out]

Piecewise((x*asinh(a*x) - sqrt(a**2*x**2 + 1)/a, Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \text {arcsinh}(a x) \, dx=\frac {a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \text {arcsinh}(a x) \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a} \]

[In]

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

[Out]

x*log(a*x + sqrt(a^2*x^2 + 1)) - sqrt(a^2*x^2 + 1)/a

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \text {arcsinh}(a x) \, dx=x\,\mathrm {asinh}\left (a\,x\right )-\frac {\sqrt {a^2\,x^2+1}}{a} \]

[In]

int(asinh(a*x),x)

[Out]

x*asinh(a*x) - (a^2*x^2 + 1)^(1/2)/a