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

Optimal. Leaf size=40 \[ -\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0710817, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5677, 5675} \[ -\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5677

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

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

Mathematica [A]  time = 0.0365253, size = 40, normalized size = 1. \[ -\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 36, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{{a}^{2}{x}^{2}+1}}{\sqrt{{\it Arcsinh} \left ( ax \right ) }a\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.58406, size = 131, normalized size = 3.28 \begin{align*} -\frac{2 \, \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} + a c\right )} \sqrt{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{asinh}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(c*(a**2*x**2 + 1))*asinh(a*x)**(3/2)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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