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 = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.89, size = 11, normalized size = 1.00 \[ \frac {\sin \relax (x)}{a \cos \relax (x) + a} \]

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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giac [A]  time = 0.35, size = 8, normalized size = 0.73 \[ \frac {\tan \left (\frac {1}{2} \, x\right )}{a} \]

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

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maple [A]  time = 0.02, size = 9, normalized size = 0.82 \[ \frac {\tan \left (\frac {x}{2}\right )}{a} \]

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 = 0.68, size = 12, normalized size = 1.09 \[ \frac {\sin \relax (x)}{a {\left (\cos \relax (x) + 1\right )}} \]

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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mupad [B]  time = 0.29, size = 8, normalized size = 0.73 \[ \frac {\mathrm {tan}\left (\frac {x}{2}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*cos(x)),x)

[Out]

tan(x/2)/a

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sympy [A]  time = 0.18, size = 5, normalized size = 0.45 \[ \frac {\tan {\left (\frac {x}{2} \right )}}{a} \]

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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