3.10 \(\int \frac{\sin (x)}{(3+\cos (x))^2} \, dx\)

Optimal. Leaf size=6 \[ \frac{1}{\cos (x)+3} \]

[Out]

(3 + Cos[x])^(-1)

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Rubi [A]  time = 0.0176402, antiderivative size = 6, normalized size of antiderivative = 1., 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 = {2668, 32} \[ \frac{1}{\cos (x)+3} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(3 + Cos[x])^2,x]

[Out]

(3 + Cos[x])^(-1)

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{(3+\cos (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(3+x)^2} \, dx,x,\cos (x)\right )\\ &=\frac{1}{3+\cos (x)}\\ \end{align*}

Mathematica [A]  time = 0.0131301, size = 6, normalized size = 1. \[ \frac{1}{\cos (x)+3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(3 + Cos[x])^2,x]

[Out]

(3 + Cos[x])^(-1)

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Maple [A]  time = 0.027, size = 7, normalized size = 1.2 \begin{align*} \left ( 3+\cos \left ( x \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(3+cos(x))^2,x)

[Out]

1/(3+cos(x))

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Maxima [A]  time = 0.945644, size = 8, normalized size = 1.33 \begin{align*} \frac{1}{\cos \left (x\right ) + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3+cos(x))^2,x, algorithm="maxima")

[Out]

1/(cos(x) + 3)

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Fricas [A]  time = 0.432652, size = 22, normalized size = 3.67 \begin{align*} \frac{1}{\cos \left (x\right ) + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3+cos(x))^2,x, algorithm="fricas")

[Out]

1/(cos(x) + 3)

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Sympy [A]  time = 0.413677, size = 5, normalized size = 0.83 \begin{align*} \frac{1}{\cos{\left (x \right )} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3+cos(x))**2,x)

[Out]

1/(cos(x) + 3)

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Giac [A]  time = 1.07614, size = 8, normalized size = 1.33 \begin{align*} \frac{1}{\cos \left (x\right ) + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3+cos(x))^2,x, algorithm="giac")

[Out]

1/(cos(x) + 3)