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

Optimal. Leaf size=47 \[ \frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n} \]

[Out]

(a+b*x^n)*arcsin(a+b*x^n)/b/n+(1-(a+b*x^n)^2)^(1/2)/b/n

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6715, 4803, 4619, 261} \[ \frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*ArcSin[a + b*x^n],x]

[Out]

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4803

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int x^{-1+n} \sin ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \sin ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n}-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 47, normalized size = 1.00 \[ \frac {\sqrt {1-\left (a+b x^n\right )^2}}{b n}+\frac {\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*ArcSin[a + b*x^n],x]

[Out]

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

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fricas [A]  time = 0.49, size = 57, normalized size = 1.21 \[ \frac {b x^{n} \arcsin \left (b x^{n} + a\right ) + a \arcsin \left (b x^{n} + a\right ) + \sqrt {-b^{2} x^{2 \, n} - 2 \, a b x^{n} - a^{2} + 1}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arcsin(a+b*x^n),x, algorithm="fricas")

[Out]

(b*x^n*arcsin(b*x^n + a) + a*arcsin(b*x^n + a) + sqrt(-b^2*x^(2*n) - 2*a*b*x^n - a^2 + 1))/(b*n)

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giac [A]  time = 0.24, size = 39, normalized size = 0.83 \[ \frac {{\left (b x^{n} + a\right )} \arcsin \left (b x^{n} + a\right ) + \sqrt {-{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arcsin(a+b*x^n),x, algorithm="giac")

[Out]

((b*x^n + a)*arcsin(b*x^n + a) + sqrt(-(b*x^n + a)^2 + 1))/(b*n)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{n -1} \arcsin \left (a +b \,x^{n}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n-1)*arcsin(a+b*x^n),x)

[Out]

int(x^(n-1)*arcsin(a+b*x^n),x)

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maxima [A]  time = 0.42, size = 39, normalized size = 0.83 \[ \frac {{\left (b x^{n} + a\right )} \arcsin \left (b x^{n} + a\right ) + \sqrt {-{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arcsin(a+b*x^n),x, algorithm="maxima")

[Out]

((b*x^n + a)*arcsin(b*x^n + a) + sqrt(-(b*x^n + a)^2 + 1))/(b*n)

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mupad [B]  time = 0.35, size = 109, normalized size = 2.32 \[ \frac {x^n\,\mathrm {asin}\left (a+b\,x^n\right )}{n}+\frac {\sqrt {1-b^2\,x^{2\,n}-2\,a\,b\,x^n-a^2}}{b\,n}+\frac {a\,\ln \left (\sqrt {1-b^2\,x^{2\,n}-2\,a\,b\,x^n-a^2}-\frac {a\,b+b^2\,x^n}{\sqrt {-b^2}}\right )}{n\,\sqrt {-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)*asin(a + b*x^n),x)

[Out]

(x^n*asin(a + b*x^n))/n + (1 - b^2*x^(2*n) - 2*a*b*x^n - a^2)^(1/2)/(b*n) + (a*log((1 - b^2*x^(2*n) - 2*a*b*x^
n - a^2)^(1/2) - (a*b + b^2*x^n)/(-b^2)^(1/2)))/(n*(-b^2)^(1/2))

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sympy [A]  time = 54.60, size = 76, normalized size = 1.62 \[ \begin {cases} \log {\relax (x )} \operatorname {asin}{\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\relax (x )} \operatorname {asin}{\left (a + b \right )} & \text {for}\: n = 0 \\\frac {x^{n} \operatorname {asin}{\relax (a )}}{n} & \text {for}\: b = 0 \\\frac {a \operatorname {asin}{\left (a + b x^{n} \right )}}{b n} + \frac {x^{n} \operatorname {asin}{\left (a + b x^{n} \right )}}{n} + \frac {\sqrt {- a^{2} - 2 a b x^{n} - b^{2} x^{2 n} + 1}}{b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*asin(a+b*x**n),x)

[Out]

Piecewise((log(x)*asin(a), Eq(b, 0) & Eq(n, 0)), (log(x)*asin(a + b), Eq(n, 0)), (x**n*asin(a)/n, Eq(b, 0)), (
a*asin(a + b*x**n)/(b*n) + x**n*asin(a + b*x**n)/n + sqrt(-a**2 - 2*a*b*x**n - b**2*x**(2*n) + 1)/(b*n), True)
)

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