\(\int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx\) [114]

Optimal result

Integrand size = 21, antiderivative size = 49 \[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {x^2}{4 a}+\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a^2}-\frac {\text {arcsinh}(a x)^2}{4 a^3} \]

[Out]

-1/4*x^2/a-1/4*arcsinh(a*x)^2/a^3+1/2*x*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 49, 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 = {5812, 5783, 30} \[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a} \]

[In]

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

[Out]

-1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5783

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

Rule 5812

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a^2}-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}-\frac {\int x \, dx}{2 a} \\ & = -\frac {x^2}{4 a}+\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a^2}-\frac {\text {arcsinh}(a x)^2}{4 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a^2 x^2-2 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\text {arcsinh}(a x)^2}{4 a^3} \]

[In]

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

[Out]

-1/4*(a^2*x^2 - 2*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]^2)/a^3

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {-2 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +a^{2} x^{2}+\operatorname {arcsinh}\left (a x \right )^{2}+1}{4 a^{3}}\)	\(40\)

[In]

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

[Out]

-1/4*(-2*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x+a^2*x^2+arcsinh(a*x)^2+1)/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a^{2} x^{2} - 2 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{3}} \]

[In]

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

[Out]

-1/4*(a^2*x^2 - 2*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1)) + log(a*x + sqrt(a^2*x^2 + 1))^2)/a^3

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{2}}{4 a} + \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{2 a^{2}} - \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{4 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-x**2/(4*a) + x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(2*a**2) - asinh(a*x)**2/(4*a**3), Ne(a, 0)), (0, Tr
ue))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {1}{4} \, a {\left (\frac {x^{2}}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )^{2}}{a^{4}}\right )} + \frac {1}{2} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )} \operatorname {arsinh}\left (a x\right ) \]

[In]

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

[Out]

-1/4*a*(x^2/a^2 - arcsinh(a*x)^2/a^4) + 1/2*(sqrt(a^2*x^2 + 1)*x/a^2 - arcsinh(a*x)/a^3)*arcsinh(a*x)

Giac [F]

\[ \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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