\(\int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx\) [180]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5798, 8} \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )+\operatorname {arcsinh}\left (a x \right )-a x \sqrt {a^{2} x^{2}+1}}{a^{2} \sqrt {a^{2} x^{2}+1}}\)	\(47\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a x - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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