3.1.27 \(\int \frac {\sin (x)}{a+b \cos (x)} \, dx\) [27]

Optimal. Leaf size=12 \[ -\frac {\log (a+b \cos (x))}{b} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 12, 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 = {2747, 31} \begin {gather*} -\frac {\log (a+b \cos (x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + b*Cos[x]),x]

[Out]

-(Log[a + b*Cos[x]]/b)

Rule 31

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

Rule 2747

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

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Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log (a+b \cos (x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + b*Cos[x]),x]

[Out]

-(Log[a + b*Cos[x]]/b)

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Maple [A]
time = 0.04, size = 13, normalized size = 1.08

method	result	size
derivativedivides	\(-\frac {\ln \left (a +b \cos \left (x \right )\right )}{b}\)	\(13\)
default	\(-\frac {\ln \left (a +b \cos \left (x \right )\right )}{b}\)	\(13\)
risch	\(\frac {i x}{b}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{b}\)	\(33\)
norman	\(\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{b}-\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{b}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b \cos \left (x\right ) + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(b*cos(x) + a)/b

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Fricas [A]
time = 0.37, size = 15, normalized size = 1.25 \begin {gather*} -\frac {\log \left (-b \cos \left (x\right ) - a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-b*cos(x) - a)/b

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Sympy [A]
time = 0.16, size = 17, normalized size = 1.42 \begin {gather*} \begin {cases} - \frac {\log {\left (\frac {a}{b} + \cos {\left (x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cos {\left (x \right )}}{a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-log(a/b + cos(x))/b, Ne(b, 0)), (-cos(x)/a, True))

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Giac [A]
time = 0.50, size = 13, normalized size = 1.08 \begin {gather*} -\frac {\log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(b*cos(x) + a))/b

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Mupad [B]
time = 0.04, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (a+b\,\cos \left (x\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a + b*cos(x)),x)

[Out]

-log(a + b*cos(x))/b

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