3.310 \(\int \sinh ^{-1}(a x^n) \, dx\)

Optimal. Leaf size=56 \[ x \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2 n},\frac{1}{2} \left (\frac{1}{n}+3\right ),-a^2 x^{2 n}\right )}{n+1} \]

[Out]

x*ArcSinh[a*x^n] - (a*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (3 + n^(-1))/2, -(a^2*x^(2*n))])/(1 +
n)

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Rubi [A]  time = 0.0225214, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5900, 12, 364} \[ x \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-a^2 x^{2 n}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x^n],x]

[Out]

x*ArcSinh[a*x^n] - (a*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (3 + n^(-1))/2, -(a^2*x^(2*n))])/(1 +
n)

Rule 5900

Int[ArcSinh[u_], x_Symbol] :> Simp[x*ArcSinh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 + u^2], x], x]
/; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \sinh ^{-1}\left (a x^n\right ) \, dx &=x \sinh ^{-1}\left (a x^n\right )-\int \frac{a n x^n}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=x \sinh ^{-1}\left (a x^n\right )-(a n) \int \frac{x^n}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=x \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{1+n} \, _2F_1\left (\frac{1}{2},\frac{1+n}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-a^2 x^{2 n}\right )}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.0249703, size = 56, normalized size = 1. \[ x \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2 n},\frac{1}{2} \left (\frac{1}{n}+3\right ),-a^2 x^{2 n}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x^n],x]

[Out]

x*ArcSinh[a*x^n] - (a*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (3 + n^(-1))/2, -(a^2*x^(2*n))])/(1 +
n)

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Maple [F]  time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{\it Arcsinh} \left ( a{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x^n),x)

[Out]

int(arcsinh(a*x^n),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^n),x, algorithm="maxima")

[Out]

-a*n*integrate(x^n/(a^3*x^(3*n) + a*x^n + (a^2*x^(2*n) + 1)^(3/2)), x) - n*x + n*integrate(1/(a^2*x^(2*n) + 1)
, x) + x*log(a*x^n + sqrt(a^2*x^(2*n) + 1))

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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^n),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x**n),x)

[Out]

Integral(asinh(a*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^n),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x^n), x)