\(\int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\) [286]

Optimal result

Integrand size = 20, antiderivative size = 13 \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^3}{3 a} \]

[Out]

1/3*arcsinh(a*x)^3/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5783} \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^3}{3 a} \]

[In]

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

[Out]

ArcSinh[a*x]^3/(3*a)

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {arcsinh}(a x)^3}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\text {arcsinh}(a x)^3}{3 a} \]

[In]

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

[Out]

ArcSinh[a*x]^3/(3*a)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{3 a}\)	\(12\)
default	\(\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{3 a}\)	\(12\)

[In]

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

[Out]

1/3*arcsinh(a*x)^3/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77 \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3}}{3 \, a} \]

[In]

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

[Out]

1/3*log(a*x + sqrt(a^2*x^2 + 1))^3/a

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{3 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((asinh(a*x)**3/(3*a), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {\operatorname {arsinh}\left (a x\right )^{3}}{3 \, a} \]

[In]

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

[Out]

1/3*arcsinh(a*x)^3/a

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx=\frac {{\mathrm {asinh}\left (a\,x\right )}^3}{3\,a} \]

[In]

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

[Out]

asinh(a*x)^3/(3*a)