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

Optimal. Leaf size=17 \[ \frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

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Rubi [A]  time = 0.0380282, antiderivative size = 17, 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} \[ \frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

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

Mathematica [A]  time = 0.0098283, size = 17, normalized size = 1. \[ \frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[a*x]^(1 + n)/(a*(1 + n))

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Maple [A]  time = 0.005, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{1+n}}{a \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(a*x)^(1+n)/a/(1+n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.52692, size = 212, normalized size = 12.47 \begin{align*} \frac{\cosh \left (n \log \left (\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )\right )\right ) \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) \sinh \left (n \log \left (\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )\right )\right )}{a n + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cosh(n*log(log(a*x + sqrt(a^2*x^2 + 1))))*log(a*x + sqrt(a^2*x^2 + 1)) + log(a*x + sqrt(a^2*x^2 + 1))*sinh(n*
log(log(a*x + sqrt(a^2*x^2 + 1)))))/(a*n + a)

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Sympy [A]  time = 1.05296, size = 34, normalized size = 2. \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = 0 \wedge n = -1 \\0^{n} x & \text{for}\: a = 0 \\\frac{\log{\left (\operatorname{asinh}{\left (a x \right )} \right )}}{a} & \text{for}\: n = -1 \\\frac{\operatorname{asinh}{\left (a x \right )} \operatorname{asinh}^{n}{\left (a x \right )}}{a n + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(n, -1)), (0**n*x, Eq(a, 0)), (log(asinh(a*x))/a, Eq(n, -1)), (asinh(a*x)*asinh
(a*x)**n/(a*n + a), True))

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Giac [A]  time = 1.39808, size = 39, normalized size = 2.29 \begin{align*} \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{n + 1}}{a{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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