\(\int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\) [285]

Optimal result

Integrand size = 21, antiderivative size = 52 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2}}{a^2}-\frac {2 x \text {arcsinh}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5798, 5772, 267} \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}+\frac {2 \sqrt {a^2 x^2+1}}{a^2}-\frac {2 x \text {arcsinh}(a x)}{a} \]

[In]

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

[Out]

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

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]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2}-2 a x \text {arcsinh}(a x)+\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.23

method	result	size
default	\(\frac {\operatorname {arcsinh}\left (a x \right )^{2} a^{2} x^{2}+\operatorname {arcsinh}\left (a x \right )^{2}-2 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +2 a^{2} x^{2}+2}{a^{2} \sqrt {a^{2} x^{2}+1}}\)	\(64\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.35 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=-\frac {2 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.94 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {2 x \operatorname {asinh}{\left (a x \right )}}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{a^{2}} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{2}}{a^{2}} - \frac {2 \, {\left (a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}\right )}}{a^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42 \[ \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{2}} - \frac {2 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a}\right )}}{a} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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