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

Optimal. Leaf size=65 \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+2}{2 n},\frac{1}{2} \left (\frac{2}{n}+3\right ),-a^2 x^{2 n}\right )}{2 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.0297645, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5902, 12, 364} \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2 n};\frac{1}{2} \left (3+\frac{2}{n}\right );-a^2 x^{2 n}\right )}{2 (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5902

Int[((a_.) + ArcSinh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 + u^2],
x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)
^(m + 1), 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 x \sinh ^{-1}\left (a x^n\right ) \, dx &=\frac{1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac{1}{2} \int \frac{a n x^{1+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=\frac{1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac{1}{2} (a n) \int \frac{x^{1+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=\frac{1}{2} x^2 \sinh ^{-1}\left (a x^n\right )-\frac{a n x^{2+n} \, _2F_1\left (\frac{1}{2},\frac{2+n}{2 n};\frac{1}{2} \left (3+\frac{2}{n}\right );-a^2 x^{2 n}\right )}{2 (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.0458168, size = 58, normalized size = 0.89 \[ \frac{x^2 \left ((n+2) \sinh ^{-1}\left (a x^n\right )-a n x^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{n}+\frac{1}{2},\frac{1}{n}+\frac{3}{2},-a^2 x^{2 n}\right )\right )}{2 (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.01, size = 0, normalized size = 0. \begin{align*} \int x{\it Arcsinh} \left ( a{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, n x^{2} - a n \int \frac{x x^{n}}{2 \,{\left (a^{3} x^{3 \, n} + a x^{n} +{\left (a^{2} x^{2 \, n} + 1\right )}^{\frac{3}{2}}\right )}}\,{d x} + \frac{1}{2} \, x^{2} \log \left (a x^{n} + \sqrt{a^{2} x^{2 \, n} + 1}\right ) + n \int \frac{x}{2 \,{\left (a^{2} x^{2 \, n} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*n*x^2 - a*n*integrate(1/2*x*x^n/(a^3*x^(3*n) + a*x^n + (a^2*x^(2*n) + 1)^(3/2)), x) + 1/2*x^2*log(a*x^n +
sqrt(a^2*x^(2*n) + 1)) + n*integrate(1/2*x/(a^2*x^(2*n) + 1), 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(x*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 x \operatorname{asinh}{\left (a x^{n} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*asinh(a*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*arcsinh(a*x^n), x)