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

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

[Out]

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

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Rubi [A]  time = 0.021915, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {5675} \[ \frac{\sinh ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]/Sqrt[1 + a^2*x^2],x]

[Out]

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

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

Mathematica [A]  time = 0.0062127, size = 13, normalized size = 1. \[ \frac{\sinh ^{-1}(a x)^2}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08765, size = 15, normalized size = 1.15 \begin{align*} \frac{\operatorname{arsinh}\left (a x\right )^{2}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.47326, size = 51, normalized size = 3.92 \begin{align*} \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x + sqrt(a^2*x^2 + 1))^2/a

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Sympy [A]  time = 0.409135, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{2 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((asinh(a*x)**2/(2*a), Ne(a, 0)), (0, True))

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Giac [B]  time = 1.39752, size = 31, normalized size = 2.38 \begin{align*} \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x + sqrt(a^2*x^2 + 1))^2/a