3.4.84 \(\int \frac {a+b \text {ArcSin}(c x^n)}{x^2} \, dx\) [384]

Optimal. Leaf size=69 \[ -\frac {a+b \text {ArcSin}\left (c x^n\right )}{x}-\frac {b c n x^{-1+n} \, _2F_1\left (\frac {1}{2},-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {4926, 12, 371} \begin {gather*} -\frac {a+b \text {ArcSin}\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} \end {gather*}

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, -1/2*(1 - n)/n, (3 - n^(-1))/2, c^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 4926

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]

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*}

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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.99 \begin {gather*} -\frac {a}{x}-\frac {b \text {ArcSin}\left (c x^n\right )}{x}+\frac {b c n x^{-1+n} \, _2F_1\left (\frac {1}{2},\frac {-1+n}{2 n};1+\frac {-1+n}{2 n};c^2 x^{2 n}\right )}{-1+n} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a +b \arcsin \left (c \,x^{n}\right )}{x^{2}}\, dx\]

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

[Out]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [C] Result contains complex when optimal does not.
time = 2.63, size = 60, normalized size = 0.87 \begin {gather*} - \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 {gather*}

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),), 1/(c**2*x**(2*n)))/(2*x*ga
mma(1 - 1/(2*n)))

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x^n\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x^n))/x^2,x)

[Out]

int((a + b*asin(c*x^n))/x^2, x)

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