3.461 \(\int \frac{1}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{2 a \sinh ^{-1}(a x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0354877, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {5675} \[ -\frac{1}{2 a \sinh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5675

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \sinh ^{-1}(a x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0073274, size = 13, normalized size = 1. \[ -\frac{1}{2 a \sinh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.32081, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2 \, a \operatorname{arsinh}\left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [B]  time = 2.6282, size = 55, normalized size = 4.23 \begin{align*} -\frac{1}{2 \, a \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 1.25273, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 a \operatorname{asinh}^{2}{\left (a x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.26494, size = 31, normalized size = 2.38 \begin{align*} -\frac{1}{2 \, a \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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