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

Optimal. Leaf size=48 \[ \frac{\left (a+b x^n\right ) \cos ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{1-\left (a+b x^n\right )^2}}{b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0544105, antiderivative size = 48, normalized size of antiderivative = 1., 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, 4804, 4620, 261} \[ \frac{\left (a+b x^n\right ) \cos ^{-1}\left (a+b x^n\right )}{b n}-\frac{\sqrt{1-\left (a+b x^n\right )^2}}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*ArcCos[a + b*x^n],x]

[Out]

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

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]

Rule 4804

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

Rule 4620

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0413817, size = 43, normalized size = 0.9 \[ \frac{\left (a+b x^n\right ) \cos ^{-1}\left (a+b x^n\right )-\sqrt{1-\left (a+b x^n\right )^2}}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}\arccos \left ( a+b{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.60569, size = 55, normalized size = 1.15 \begin{align*} \frac{{\left (b x^{n} + a\right )} \arccos \left (b x^{n} + a\right ) - \sqrt{-{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.64617, size = 132, normalized size = 2.75 \begin{align*} \frac{b x^{n} \arccos \left (b x^{n} + a\right ) + a \arccos \left (b x^{n} + a\right ) - \sqrt{-b^{2} x^{2 \, n} - 2 \, a b x^{n} - a^{2} + 1}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.31185, size = 55, normalized size = 1.15 \begin{align*} \frac{{\left (b x^{n} + a\right )} \arccos \left (b x^{n} + a\right ) - \sqrt{-{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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