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

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

[Out]

-(ArcSinh[a*x^n]/x) - (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 = 0.0331579, antiderivative size = 65, 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} \[ -\frac{a n x^{n-1} \, _2F_1\left (\frac{1}{2},-\frac{1-n}{2 n};\frac{1}{2} \left (3-\frac{1}{n}\right );-a^2 x^{2 n}\right )}{1-n}-\frac{\sinh ^{-1}\left (a x^n\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcSinh[a*x^n]/x) - (a*n*x^(-1 + n)*Hypergeometric2F1[1/2, -(1 - n)/(2*n), (3 - n^(-1))/2, -(a^2*x^(2*n))])/
(1 - 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^2} \, dx &=-\frac{\sinh ^{-1}\left (a x^n\right )}{x}+\int \frac{a n x^{-2+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=-\frac{\sinh ^{-1}\left (a x^n\right )}{x}+(a n) \int \frac{x^{-2+n}}{\sqrt{1+a^2 x^{2 n}}} \, dx\\ &=-\frac{\sinh ^{-1}\left (a x^n\right )}{x}-\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 = 0.0526311, size = 61, normalized size = 0.94 \[ \frac{a n x^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-1}{2 n},\frac{n-1}{2 n}+1,-a^2 x^{2 n}\right )}{n-1}-\frac{\sinh ^{-1}\left (a x^n\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Arcsinh} \left ( a{x}^{n} \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arcsinh(a*x^n)/x^2,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^{2} x^{3 \, n} + a x^{2} x^{n} +{\left (a^{2} x^{2} x^{2 \, n} + x^{2}\right )} \sqrt{a^{2} x^{2 \, n} + 1}}\,{d x} - n \int \frac{1}{a^{2} x^{2} x^{2 \, n} + x^{2}}\,{d x} - \frac{n + \log \left (a x^{n} + \sqrt{a^{2} x^{2 \, n} + 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*n*integrate(x^n/(a^3*x^2*x^(3*n) + a*x^2*x^n + (a^2*x^2*x^(2*n) + x^2)*sqrt(a^2*x^(2*n) + 1)), x) - n*integr
ate(1/(a^2*x^2*x^(2*n) + x^2), x) - (n + log(a*x^n + sqrt(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(arcsinh(a*x^n)/x^2,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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asinh(a*x**n)/x**2, 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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x^n)/x^2, x)