3.4 \(\int \frac{\sin (x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=10 \[ -\frac{\log (\cos (x)+1)}{a} \]

[Out]

-(Log[1 + Cos[x]]/a)

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Rubi [A]  time = 0.0235585, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2667, 31} \[ -\frac{\log (\cos (x)+1)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + a*Cos[x]),x]

[Out]

-(Log[1 + Cos[x]]/a)

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])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0058053, size = 12, normalized size = 1.2 \[ -\frac{2 \log \left (\cos \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + a*Cos[x]),x]

[Out]

(-2*Log[Cos[x/2]])/a

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Maple [A]  time = 0.031, size = 13, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( a+a\cos \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a+a*cos(x)),x)

[Out]

-ln(a+a*cos(x))/a

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Maxima [A]  time = 1.11335, size = 16, normalized size = 1.6 \begin{align*} -\frac{\log \left (a \cos \left (x\right ) + a\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*cos(x)),x, algorithm="maxima")

[Out]

-log(a*cos(x) + a)/a

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Fricas [A]  time = 1.61993, size = 35, normalized size = 3.5 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*cos(x)),x, algorithm="fricas")

[Out]

-log(1/2*cos(x) + 1/2)/a

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Sympy [A]  time = 0.146093, size = 8, normalized size = 0.8 \begin{align*} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*cos(x)),x)

[Out]

-log(cos(x) + 1)/a

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Giac [A]  time = 1.10571, size = 14, normalized size = 1.4 \begin{align*} -\frac{\log \left (\cos \left (x\right ) + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(a+a*cos(x)),x, algorithm="giac")

[Out]

-log(cos(x) + 1)/a