\(\int \sqrt {1+x^2} \text {arcsinh}(x) \, dx\) [79]

Optimal result

Integrand size = 12, antiderivative size = 32 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x \sqrt {1+x^2} \text {arcsinh}(x)+\frac {\text {arcsinh}(x)^2}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5785, 5783, 30} \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\frac {1}{2} \sqrt {x^2+1} x \text {arcsinh}(x)+\frac {\text {arcsinh}(x)^2}{4}-\frac {x^2}{4} \]

[In]

Int[Sqrt[1 + x^2]*ArcSinh[x],x]

[Out]

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

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 5785

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\frac {1}{4} \left (-x^2+2 x \sqrt {1+x^2} \text {arcsinh}(x)+\text {arcsinh}(x)^2\right ) \]

[In]

Integrate[Sqrt[1 + x^2]*ArcSinh[x],x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {x \,\operatorname {arcsinh}\left (x \right ) \sqrt {x^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (x \right )^{2}}{4}-\frac {x^{2}}{4}-\frac {1}{4}\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=\frac {1}{2} \, \sqrt {x^{2} + 1} x \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \log \left (x + \sqrt {x^{2} + 1}\right )^{2} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(x**2 + 1)*asinh(x), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \sqrt {1+x^2} \text {arcsinh}(x) \, dx=-\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {x^{2} + 1} x + \operatorname {arsinh}\left (x\right )\right )} \operatorname {arsinh}\left (x\right ) - \frac {1}{4} \, \operatorname {arsinh}\left (x\right )^{2} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(asinh(x)*(x^2 + 1)^(1/2),x)

[Out]

int(asinh(x)*(x^2 + 1)^(1/2), x)