3.12 \(\int \frac {\sin (x)}{(1+\cos (x))^2} \, dx\)

Optimal. Leaf size=6 \[ \frac {1}{\cos (x)+1} \]

[Out]

1/(1+cos(x))

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Rubi [A]  time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2667, 32} \[ \frac {1}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(1 + Cos[x])^2,x]

[Out]

(1 + Cos[x])^(-1)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=\frac {1}{1+\cos (x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 2.00 \[ \frac {1}{2} \sec ^2\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(1 + Cos[x])^2,x]

[Out]

Sec[x/2]^2/2

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fricas [A]  time = 0.98, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))^2,x, algorithm="fricas")

[Out]

1/(cos(x) + 1)

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giac [A]  time = 0.48, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))^2,x, algorithm="giac")

[Out]

1/(cos(x) + 1)

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maple [A]  time = 0.02, size = 7, normalized size = 1.17 \[ \frac {1}{\cos \relax (x )+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x)+1)^2,x)

[Out]

1/(cos(x)+1)

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maxima [A]  time = 0.60, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))^2,x, algorithm="maxima")

[Out]

1/(cos(x) + 1)

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mupad [B]  time = 0.25, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x)+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x) + 1)^2,x)

[Out]

1/(cos(x) + 1)

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sympy [A]  time = 0.32, size = 5, normalized size = 0.83 \[ \frac {1}{\cos {\relax (x )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x))**2,x)

[Out]

1/(cos(x) + 1)

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