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

Optimal. Leaf size=77 \[ \frac{x^{m+1} \sinh ^{-1}\left (a x^n\right )}{m+1}-\frac{a n x^{m+n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+n+1}{2 n},\frac{m+3 n+1}{2 n},-a^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.044325, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5902, 12, 364} \[ \frac{x^{m+1} \sinh ^{-1}\left (a x^n\right )}{m+1}-\frac{a n x^{m+n+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};-a^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*ArcSinh[a*x^n])/(1 + m) - (a*n*x^(1 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n
)/(2*n), -(a^2*x^(2*n))])/((1 + m)*(1 + m + 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^m \sinh ^{-1}\left (a x^n\right ) \, dx &=\frac{x^{1+m} \sinh ^{-1}\left (a x^n\right )}{1+m}-\frac{\int \frac{a n x^{m+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx}{1+m}\\ &=\frac{x^{1+m} \sinh ^{-1}\left (a x^n\right )}{1+m}-\frac{(a n) \int \frac{x^{m+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx}{1+m}\\ &=\frac{x^{1+m} \sinh ^{-1}\left (a x^n\right )}{1+m}-\frac{a n x^{1+m+n} \, _2F_1\left (\frac{1}{2},\frac{1+m+n}{2 n};\frac{1+m+3 n}{2 n};-a^2 x^{2 n}\right )}{(1+m) (1+m+n)}\\ \end{align*}

Mathematica [A]  time = 0.0738982, size = 74, normalized size = 0.96 \[ \frac{x^{m+1} \left ((m+n+1) \sinh ^{-1}\left (a x^n\right )-a n x^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+n+1}{2 n},\frac{m+3 n+1}{2 n},-a^2 x^{2 n}\right )\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}{\it Arcsinh} \left ( a{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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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^m*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^{m} \operatorname{asinh}{\left (a x^{n} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*asinh(a*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*arcsinh(a*x^n), x)