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

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

[Out]

1/(3+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 = {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 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 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]

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*}

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Mathematica [A]  time = 0.02, size = 6, normalized size = 1.00 \[ \frac {1}{\cos (x)+3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(3 + Cos[x])^2,x]

[Out]

(3 + Cos[x])^(-1)

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fricas [A]  time = 0.41, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 3} \]

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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giac [A]  time = 0.01, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 3} \]

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)

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maple [A]  time = 0.07, size = 7, normalized size = 1.17 \[ \frac {1}{\cos \relax (x )+3} \]

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 = 0.70, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 3} \]

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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mupad [B]  time = 0.04, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x)+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x) + 3)^2,x)

[Out]

1/(cos(x) + 3)

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sympy [A]  time = 0.41, size = 5, normalized size = 0.83 \[ \frac {1}{\cos {\relax (x )} + 3} \]

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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