3.4.13 \(\int \frac {\sinh ^{-1}(a x^n)}{x^3} \, dx\) [313]

Optimal. Leaf size=68 \[ -\frac {\sinh ^{-1}\left (a x^n\right )}{2 x^2}-\frac {a n x^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right );\frac {1}{2} \left (3-\frac {2}{n}\right );-a^2 x^{2 n}\right )}{2 (2-n)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 68, 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-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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*ArcSinh[a*x^n]/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 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^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*}

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Mathematica [A]
time = 0.03, size = 62, normalized size = 0.91 \begin {gather*} \frac {-\left ((-2+n) \sinh ^{-1}\left (a x^n\right )\right )+a n x^n \, _2F_1\left (\frac {1}{2},\frac {1}{2}-\frac {1}{n};\frac {3}{2}-\frac {1}{n};-a^2 x^{2 n}\right )}{2 (-2+n) x^2} \end {gather*}

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 = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\arcsinh \left (a \,x^{n}\right )}{x^{3}}\, dx\]

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 = 0.00, 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^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 = 0.00, 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^3,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^{3}}\, dx \end {gather*}

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 = 0.00, 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^3,x, algorithm="giac")

[Out]

integrate(arcsinh(a*x^n)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(asinh(a*x^n)/x^3, x)

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