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

Optimal. Leaf size=60 \[ -\frac{b c n x^{n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2 n},\frac{1}{2} \left (\frac{1}{n}+3\right ),c^2 x^{2 n}\right )}{n+1}+a x+b x \sin ^{-1}\left (c x^n\right ) \]

[Out]

a*x + b*x*ArcSin[c*x^n] - (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 = 0.0349295, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4840, 12, 364} \[ a x-\frac{b c n x^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );c^2 x^{2 n}\right )}{n+1}+b x \sin ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSin[c*x^n],x]

[Out]

a*x + b*x*ArcSin[c*x^n] - (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 4840

Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /;
InverseFunctionFreeQ[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 \left (a+b \sin ^{-1}\left (c x^n\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (c x^n\right ) \, dx\\ &=a x+b x \sin ^{-1}\left (c x^n\right )-b \int \frac{c n x^n}{\sqrt{1-c^2 x^{2 n}}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^n\right )-(b c n) \int \frac{x^n}{\sqrt{1-c^2 x^{2 n}}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^n\right )-\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 = 0.0328912, size = 60, normalized size = 1. \[ -\frac{b c n x^{n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2 n},\frac{1}{2} \left (\frac{1}{n}+3\right ),c^2 x^{2 n}\right )}{n+1}+a x+b x \sin ^{-1}\left (c x^n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSin[c*x^n],x]

[Out]

a*x + b*x*ArcSin[c*x^n] - (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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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int a+b\arcsin \left ( c{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*n*integrate(sqrt(c*x^n + 1)*sqrt(-c*x^n + 1)*x^n/(c^2*x^(2*n) - 1), x) + x*arctan2(c*x^n, sqrt(c*x^n + 1)*s
qrt(-c*x^n + 1)))*b + a*x

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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, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 3.22628, size = 56, normalized size = 0.93 \begin{align*} a x + b \left (x \operatorname{asin}{\left (c x^{n} \right )} + \frac{i x \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 \Gamma \left (1 + \frac{1}{2 n}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*arcsin(c*x^n) + a, x)