3.1.4 \(\int \frac {\sin (x)}{a+a \cos (x)} \, dx\) [4]

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

[Out]

-ln(cos(x)+1)/a

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 = {2746, 31} \begin {gather*} -\frac {\log (\cos (x)+1)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + a*Cos[x]),x]

[Out]

-(Log[1 + Cos[x]]/a)

Rule 31

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

Rule 2746

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

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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.20 \begin {gather*} -\frac {2 \log \left (\cos \left (\frac {x}{2}\right )\right )}{a} \end {gather*}

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 = 0.04, size = 13, normalized size = 1.30

method	result	size
derivativedivides	\(-\frac {\ln \left (a +a \cos \left (x \right )\right )}{a}\)	\(13\)
default	\(-\frac {\ln \left (a +a \cos \left (x \right )\right )}{a}\)	\(13\)
norman	\(\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{a}\)	\(14\)
risch	\(\frac {i x}{a}-\frac {2 \ln \left ({\mathrm e}^{i x}+1\right )}{a}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a+a*cos(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 12, normalized size = 1.20 \begin {gather*} -\frac {\log \left (a \cos \left (x\right ) + a\right )}{a} \end {gather*}

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 = 0.36, size = 12, normalized size = 1.20 \begin {gather*} -\frac {\log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{a} \end {gather*}

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 = 0.05, size = 8, normalized size = 0.80 \begin {gather*} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{a} \end {gather*}

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 = 0.41, size = 10, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\cos \left (x\right ) + 1\right )}{a} \end {gather*}

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

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Mupad [B]
time = 0.06, size = 10, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (\cos \left (x\right )+1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a + a*cos(x)),x)

[Out]

-log(cos(x) + 1)/a

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