3.1.9 \(\int \frac {1}{1+\cos (x)} \, dx\) [9]

Optimal. Leaf size=9 \[ \frac {\sin (x)}{1+\cos (x)} \]

[Out]

sin(x)/(cos(x)+1)

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2727} \begin {gather*} \frac {\sin (x)}{\cos (x)+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x])^(-1),x]

[Out]

Sin[x]/(1 + Cos[x])

Rule 2727

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

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Mathematica [A]
time = 0.00, size = 6, normalized size = 0.67 \begin {gather*} \tan \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x])^(-1),x]

[Out]

Tan[x/2]

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Maple [A]
time = 0.02, size = 5, normalized size = 0.56

method	result	size
default	\(\tan \left (\frac {x}{2}\right )\)	\(5\)
norman	\(\tan \left (\frac {x}{2}\right )\)	\(5\)
risch	\(\frac {2 i}{1+{\mathrm e}^{i x}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x)),x,method=_RETURNVERBOSE)

[Out]

tan(1/2*x)

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Maxima [A]
time = 2.44, size = 9, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(cos(x) + 1)

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Fricas [A]
time = 1.11, size = 9, normalized size = 1.00 \begin {gather*} \frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(cos(x) + 1)

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Sympy [A]
time = 0.08, size = 3, normalized size = 0.33 \begin {gather*} \tan {\left (\frac {x}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (9) = 18\).
time = 1.12, size = 30, normalized size = 3.33 \begin {gather*} -\frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (x^{2} + 1\right )} {\left (\frac {x^{2} - 1}{x^{2} + 1} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*tan(1/2*x)/((x^2 + 1)*((x^2 - 1)/(x^2 + 1) - 1))

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Mupad [B]
time = 0.19, size = 4, normalized size = 0.44 \begin {gather*} \mathrm {tan}\left (\frac {x}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x) + 1),x)

[Out]

tan(x/2)

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