3.385 \(\int \frac{a+b \sin ^{-1}(c x^n)}{x^3} \, dx\)

Optimal. Leaf size=72 \[ -\frac{b c n x^{n-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} \left (1-\frac{2}{n}\right ),\frac{1}{2} \left (3-\frac{2}{n}\right ),c^2 x^{2 n}\right )}{2 (2-n)}-\frac{a+b \sin ^{-1}\left (c x^n\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0453786, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4842, 12, 364} \[ -\frac{a+b \sin ^{-1}\left (c x^n\right )}{2 x^2}-\frac{b c n x^{n-2} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1-\frac{2}{n}\right );\frac{1}{2} \left (3-\frac{2}{n}\right );c^2 x^{2 n}\right )}{2 (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x^n])/x^3,x]

[Out]

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

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
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{a+b \sin ^{-1}\left (c x^n\right )}{x^3} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^n\right )}{2 x^2}+\frac{1}{2} b \int \frac{c n x^{-3+n}}{\sqrt{1-c^2 x^{2 n}}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^n\right )}{2 x^2}+\frac{1}{2} (b c n) \int \frac{x^{-3+n}}{\sqrt{1-c^2 x^{2 n}}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^n\right )}{2 x^2}-\frac{b c 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 );c^2 x^{2 n}\right )}{2 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.0526302, size = 75, normalized size = 1.04 \[ \frac{b c n x^{n-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-2}{2 n},\frac{n-2}{2 n}+1,c^2 x^{2 n}\right )}{2 (n-2)}-\frac{a}{2 x^2}-\frac{b \sin ^{-1}\left (c x^n\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x^n])/x^3,x]

[Out]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arcsin \left ( c{x}^{n} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x^n))/x^3,x)

[Out]

int((a+b*arcsin(c*x^n))/x^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^n))/x^3,x, algorithm="maxima")

[Out]

-1/2*(2*c*n*x^2*integrate(1/2*sqrt(c*x^n + 1)*sqrt(-c*x^n + 1)*x^n/(c^2*x^3*x^(2*n) - x^3), x) + arctan2(c*x^n
, sqrt(c*x^n + 1)*sqrt(-c*x^n + 1)))*b/x^2 - 1/2*a/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((a+b*arcsin(c*x^n))/x^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 23.8386, size = 61, normalized size = 0.85 \begin{align*} - \frac{a}{2 x^{2}} - \frac{b \operatorname{asin}{\left (c x^{n} \right )}}{2 x^{2}} - \frac{i b \Gamma \left (- \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{\frac{x^{- 2 n}}{c^{2}}} \right )}}{4 x^{2} \Gamma \left (1 - \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x**n))/x**3,x)

[Out]

-a/(2*x**2) - b*asin(c*x**n)/(2*x**2) - I*b*gamma(-1/n)*hyper((1/2, 1/n), (1 + 1/n,), x**(-2*n)/c**2)/(4*x**2*
gamma(1 - 1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^n))/x^3,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x^n) + a)/x^3, x)