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

Optimal. Leaf size=72 \[ -\frac {a+b \sin ^{-1}\left (c x^n\right )}{2 x^2}-\frac {b c n x^{n-2} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (1-\frac {2}{n}\right );\frac {1}{2} \left (3-\frac {2}{n}\right );c^2 x^{2 n}\right )}{2 (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 72, 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 = {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 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])

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]

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*}

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Mathematica [A]  time = 0.06, size = 75, normalized size = 1.04 \[ -\frac {a}{2 x^2}+\frac {b c n x^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2 n};\frac {n-2}{2 n}+1;c^2 x^{2 n}\right )}{2 (n-2)}-\frac {b \sin ^{-1}\left (c x^n\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*a/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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x^{n}\right ) + a}{x^{3}}\,{d x} \]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arcsin \left (c \,x^{n}\right )}{x^{3}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c n x^{2} \int \frac {\sqrt {c x^{n} + 1} \sqrt {-c x^{n} + 1} x^{n}}{c^{2} x^{2 \, n + 3} - x^{3}}\,{d x} + \arctan \left (c x^{n}, \sqrt {c x^{n} + 1} \sqrt {-c x^{n} + 1}\right )\right )} b}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^n\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x^n))/x^3,x)

[Out]

int((a + b*asin(c*x^n))/x^3, x)

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sympy [C]  time = 7.60, size = 61, normalized size = 0.85 \[ - \frac {a}{2 x^{2}} - \frac {b \operatorname {asin}{\left (c x^{n} \right )}}{2 x^{2}} - \frac {i b \Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {x^{- 2 n}}{c^{2}}} \right )}}{4 x^{2} \Gamma \left (1 - \frac {1}{n}\right )} \]

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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