3.4.12 \(\int \frac {\sinh ^{-1}(a x^n)}{x^2} \, dx\) [312]

Optimal. Leaf size=65 \[ -\frac {\sinh ^{-1}\left (a x^n\right )}{x}-\frac {a n x^{-1+n} \, _2F_1\left (\frac {1}{2},-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );-a^2 x^{2 n}\right )}{1-n} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5875, 12, 371} \begin {gather*} -\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} \end {gather*}

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, -1/2*(1 - n)/n, (3 - n^(-1))/2, -(a^2*x^(2*n))])/
(1 - n)

Rule 12

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 5875

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]

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*}

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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.94 \begin {gather*} -\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};1+\frac {-1+n}{2 n};-a^2 x^{2 n}\right )}{-1+n} \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\arcsinh \left (a \,x^{n}\right )}{x^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}{\left (a x^{n} \right )}}{x^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(asinh(a*x^n)/x^2, x)

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