3.5 \(\int \frac{1}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=11 \[ \frac{\sin (x)}{a \cos (x)+a} \]

[Out]

Sin[x]/(a + a*Cos[x])

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Rubi [A]  time = 0.0103396, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2648} \[ \frac{\sin (x)}{a \cos (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[x])^(-1),x]

[Out]

Sin[x]/(a + a*Cos[x])

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0046866, size = 10, normalized size = 0.91 \[ \frac{\tan \left (\frac{x}{2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[x])^(-1),x]

[Out]

Tan[x/2]/a

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Maple [A]  time = 0.025, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*cos(x)),x)

[Out]

1/a*tan(1/2*x)

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Maxima [A]  time = 1.10538, size = 16, normalized size = 1.45 \begin{align*} \frac{\sin \left (x\right )}{a{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(x)),x, algorithm="maxima")

[Out]

sin(x)/(a*(cos(x) + 1))

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Fricas [A]  time = 1.51081, size = 31, normalized size = 2.82 \begin{align*} \frac{\sin \left (x\right )}{a \cos \left (x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(x)),x, algorithm="fricas")

[Out]

sin(x)/(a*cos(x) + a)

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Sympy [A]  time = 0.221091, size = 5, normalized size = 0.45 \begin{align*} \frac{\tan{\left (\frac{x}{2} \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x/2)/a

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Giac [A]  time = 1.15774, size = 11, normalized size = 1. \begin{align*} \frac{\tan \left (\frac{1}{2} \, x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(x)),x, algorithm="giac")

[Out]

tan(1/2*x)/a