3.117 \(\int \frac{\cos ^{-1}(\sqrt{1+b x^2})^n}{\sqrt{1+b x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\sqrt{-b x^2} \cos ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0653671, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {4835, 4642} \[ -\frac{\sqrt{-b x^2} \cos ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]

[Out]

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

Rule 4835

Int[ArcCos[Sqrt[1 + (b_.)*(x_)^2]]^(n_.)/Sqrt[1 + (b_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[-(b*x^2)]/(b*x), Subst
[Int[ArcCos[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]], x] /; FreeQ[{b, n}, x]

Rule 4642

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

Rubi steps

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

Mathematica [A]  time = 0.0458177, size = 39, normalized size = 1. \[ -\frac{\sqrt{-b x^2} \cos ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]

[Out]

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

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Maple [F]  time = 0.193, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arccos \left ( \sqrt{b{x}^{2}+1} \right ) \right ) ^{n}{\frac{1}{\sqrt{b{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x)

[Out]

int(arccos((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.99523, size = 107, normalized size = 2.74 \begin{align*} -\frac{\sqrt{-b x^{2}} \arccos \left (\sqrt{b x^{2} + 1}\right )^{n} \arccos \left (\sqrt{b x^{2} + 1}\right )}{{\left (b n + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos((b*x**2+1)**(1/2))**n/(b*x**2+1)**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos((b*x^2+1)^(1/2))^n/(b*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arccos(sqrt(b*x^2 + 1))^n/sqrt(b*x^2 + 1), x)