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]

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

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Rubi [A]  time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.05, size = 39, normalized size = 1.00 \[ -\frac {\sqrt {-b x^2} \cos ^{-1}\left (\sqrt {b x^2+1}\right )^{n+1}}{b (n+1) x} \]

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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fricas [A]  time = 0.48, size = 42, normalized size = 1.08 \[ -\frac {\sqrt {-b x^{2}} \arccos \left (\sqrt {b x^{2} + 1}\right )^{n} \arccos \left (\sqrt {b x^{2} + 1}\right )}{{\left (b n + b\right )} x} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\arccos \left (\sqrt {b \,x^{2}+1}\right )^{n}}{\sqrt {b \,x^{2}+1}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found sqrt(-_SAGE_VAR_b)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {acos}\left (\sqrt {b\,x^2+1}\right )}^n}{\sqrt {b\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \tilde {\infty } x & \text {for}\: b = 0 \wedge n = -1 \\0^{n} x & \text {for}\: b = 0 \\\int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {acos}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx & \text {for}\: n = -1 \\- \frac {i \sqrt {b} \sqrt {x^{2}} \operatorname {acos}{\left (\sqrt {b x^{2} + 1} \right )} \operatorname {acos}^{n}{\left (\sqrt {b x^{2} + 1} \right )}}{b n x + b x} & \text {otherwise} \end {cases} \]

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]

Piecewise((zoo*x, Eq(b, 0) & Eq(n, -1)), (0**n*x, Eq(b, 0)), (Integral(1/(sqrt(b*x**2 + 1)*acos(sqrt(b*x**2 +
1))), x), Eq(n, -1)), (-I*sqrt(b)*sqrt(x**2)*acos(sqrt(b*x**2 + 1))*acos(sqrt(b*x**2 + 1))**n/(b*n*x + b*x), T
rue))

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