\(\int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx\) [461]

Optimal result

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

[Out]

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

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} \[ \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx=-\frac {1}{2 a \text {arcsinh}(a x)^2} \]

[In]

Int[1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3),x]

[Out]

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

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 {1}{2 a \text {arcsinh}(a x)^2} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3),x]

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(-\frac {1}{2 a \operatorname {arcsinh}\left (a x \right )^{2}}\)	\(12\)
default	\(-\frac {1}{2 a \operatorname {arcsinh}\left (a x \right )^{2}}\)	\(12\)

[In]

int(1/arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

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

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx=-\frac {1}{2 \, a \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}} \]

[In]

integrate(1/arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx=- \frac {1}{2 a \operatorname {asinh}^{2}{\left (a x \right )}} \]

[In]

integrate(1/asinh(a*x)**3/(a**2*x**2+1)**(1/2),x)

[Out]

-1/(2*a*asinh(a*x)**2)

Maxima [A] (verification not implemented)

none

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

[In]

integrate(1/arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a^2*x^2 + 1)*arcsinh(a*x)^3), x)

Mupad [B] (verification not implemented)

Time = 2.68 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3} \, dx=-\frac {1}{2\,a\,{\mathrm {asinh}\left (a\,x\right )}^2} \]

[In]

int(1/(asinh(a*x)^3*(a^2*x^2 + 1)^(1/2)),x)

[Out]

-1/(2*a*asinh(a*x)^2)