3.384 \(\int \frac{a+b \sin ^{-1}(c x^n)}{x^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{b c n x^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{1-n}{2 n},\frac{1}{2} \left (3-\frac{1}{n}\right ),c^2 x^{2 n}\right )}{1-n}-\frac{a+b \sin ^{-1}\left (c x^n\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.0431259, antiderivative size = 69, 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} \[ -\frac{a+b \sin ^{-1}\left (c x^n\right )}{x}-\frac{b c n x^{n-1} \, _2F_1\left (\frac{1}{2},-\frac{1-n}{2 n};\frac{1}{2} \left (3-\frac{1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x^n])/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0717266, size = 68, normalized size = 0.99 \[ \frac{b c n x^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-1}{2 n},\frac{n-1}{2 n}+1,c^2 x^{2 n}\right )}{n-1}-\frac{a}{x}-\frac{b \sin ^{-1}\left (c x^n\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arcsin \left ( c{x}^{n} \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x^n))/x^2,x)

[Out]

int((a+b*arcsin(c*x^n))/x^2,x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^n))/x^2,x, algorithm="maxima")

[Out]

Timed out

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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^2,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 8.22533, size = 60, normalized size = 0.87 \begin{align*} - \frac{a}{x} - \frac{b \operatorname{asin}{\left (c x^{n} \right )}}{x} - \frac{i b \Gamma \left (- \frac{1}{2 n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{1}{2 n} \\ 1 + \frac{1}{2 n} \end{matrix}\middle |{\frac{x^{- 2 n}}{c^{2}}} \right )}}{2 x \Gamma \left (1 - \frac{1}{2 n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x**n))/x**2,x)

[Out]

-a/x - b*asin(c*x**n)/x - I*b*gamma(-1/(2*n))*hyper((1/2, 1/(2*n)), (1 + 1/(2*n),), x**(-2*n)/c**2)/(2*x*gamma
(1 - 1/(2*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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^n))/x^2,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x^n) + a)/x^2, x)