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

Optimal. Leaf size=42 \[ \frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.0771497, antiderivative size = 42, 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} \[ \frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0378993, size = 42, normalized size = 1. \[ \frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 36, normalized size = 0.9 \begin{align*}{\frac{2}{3\,a} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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