3.4.82 \(\int (a+b \text {ArcSin}(c x^n)) \, dx\) [382]

Optimal. Leaf size=60 \[ a x+b x \text {ArcSin}\left (c x^n\right )-\frac {b c n x^{1+n} \, _2F_1\left (\frac {1}{2},\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );c^2 x^{2 n}\right )}{1+n} \]

[Out]

a*x+b*x*arcsin(c*x^n)-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 = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4924, 12, 371} \begin {gather*} a x+b x \text {ArcSin}\left (c x^n\right )-\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} \end {gather*}

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 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 4924

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]

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*}

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Mathematica [A]
time = 0.04, size = 60, normalized size = 1.00 \begin {gather*} a x+b x \text {ArcSin}\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 {gather*}

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

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 = 0.00, 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, 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 = 0.00, 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, 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 = 1.30, size = 56, normalized size = 0.93 \begin {gather*} 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 {gather*}

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),), 1/(c**2*x**(2*n)))/(2*gamm
a(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, algorithm="giac")

[Out]

integrate(b*arcsin(c*x^n) + a, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int a+b\,\mathrm {asin}\left (c\,x^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*asin(c*x^n),x)

[Out]

int(a + b*asin(c*x^n), x)

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