3.313 \(\int \frac{\sinh ^{-1}(a x^n)}{x^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{a n x^{n-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (1-\frac{2}{n}\right ),\frac{1}{2} \left (3-\frac{2}{n}\right ),-a^2 x^{2 n}\right )}{2 (2-n)}-\frac{\sinh ^{-1}\left (a x^n\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0390606, antiderivative size = 68, 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} \[ -\frac{a n x^{n-2} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1-\frac{2}{n}\right );\frac{1}{2} \left (3-\frac{2}{n}\right );-a^2 x^{2 n}\right )}{2 (2-n)}-\frac{\sinh ^{-1}\left (a x^n\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x^n]/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0436203, size = 62, normalized size = 0.91 \[ \frac{a n x^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2}-\frac{1}{n},\frac{3}{2}-\frac{1}{n},-a^2 x^{2 n}\right )-(n-2) \sinh ^{-1}\left (a x^n\right )}{2 (n-2) x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x^n]/x^3,x]

[Out]

(-((-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)*x^2)

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Maple [F]  time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Arcsinh} \left ( a{x}^{n} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x^n)/x^3,x)

[Out]

int(arcsinh(a*x^n)/x^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^n)/x^3,x, algorithm="maxima")

[Out]

a*n*integrate(1/2*x^n/(a^3*x^3*x^(3*n) + a*x^3*x^n + (a^2*x^3*x^(2*n) + x^3)*sqrt(a^2*x^(2*n) + 1)), x) - n*in
tegrate(1/2/(a^2*x^3*x^(2*n) + x^3), x) - 1/4*(n + 2*log(a*x^n + sqrt(a^2*x^(2*n) + 1)))/x^2

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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^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x^{n} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x**n)/x**3,x)

[Out]

Integral(asinh(a*x**n)/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x^{n}\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^n)/x^3,x, algorithm="giac")

[Out]

integrate(arcsinh(a*x^n)/x^3, x)