3.179 \(\int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx\)

Optimal. Leaf size=17 \[ \text{Unintegrable}\left (\frac{1}{x \sqrt{a-a \cos (x)}},x\right ) \]

[Out]

Unintegrable[1/(x*Sqrt[a - a*Cos[x]]), x]

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Rubi [A]  time = 0.0772828, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*Sqrt[a - a*Cos[x]]),x]

[Out]

Defer[Int][1/(x*Sqrt[a - a*Cos[x]]), x]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx &=\int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 3.12421, size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*Sqrt[a - a*Cos[x]]),x]

[Out]

Integrate[1/(x*Sqrt[a - a*Cos[x]]), x]

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Maple [A]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt{a-a\cos \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a-a*cos(x))^(1/2),x)

[Out]

int(1/x/(a-a*cos(x))^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a \cos \left (x\right ) + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a-a*cos(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-a*cos(x) + a)*x), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a \cos \left (x\right ) + a}}{a x \cos \left (x\right ) - a x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a-a*cos(x))^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a*cos(x) + a)/(a*x*cos(x) - a*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- a \left (\cos{\left (x \right )} - 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a-a*cos(x))**(1/2),x)

[Out]

Integral(1/(x*sqrt(-a*(cos(x) - 1))), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a \cos \left (x\right ) + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a-a*cos(x))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-a*cos(x) + a)*x), x)