3.11 \(\int \frac {\sin (x)}{a-b \cos (x)} \, dx\)

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

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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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2668, 31} \[ \frac {\log (a-b \cos (x))}{b} \]

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 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)}{a-b \cos (x)} \, dx &=\frac {\operatorname {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 \[ \frac {\log (a-b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a - b*Cos[x]),x]

[Out]

Log[a - b*Cos[x]]/b

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{a-b \cos (x)} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sin[x]/(a - b*Cos[x]),x]

[Out]

Could not integrate

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fricas [A]  time = 0.87, size = 12, normalized size = 1.00 \[ \frac {\log \left (-b \cos \relax (x) + a\right )}{b} \]

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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giac [A]  time = 1.08, size = 14, normalized size = 1.17 \[ \frac {\log \left ({\left | b \cos \relax (x) - a \right |}\right )}{b} \]

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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maple [A]  time = 0.07, size = 13, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {\ln \left (a -b \cos \relax (x )\right )}{b}\)	\(13\)
default	\(\frac {\ln \left (a -b \cos \relax (x )\right )}{b}\)	\(13\)
risch	\(-\frac {i x}{b}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{b}\)	\(32\)
norman	\(\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a -b \right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b}\)	\(42\)

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 = 0.44, size = 13, normalized size = 1.08 \[ \frac {\log \left (b \cos \relax (x) - a\right )}{b} \]

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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mupad [B]  time = 0.20, size = 13, normalized size = 1.08 \[ \frac {\ln \left (b\,\cos \relax (x)-a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(a - b*cos(x)),x)

[Out]

log(b*cos(x) - a)/b

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sympy [A]  time = 0.37, size = 15, normalized size = 1.25 \[ \begin {cases} \frac {\log {\left (- \frac {a}{b} + \cos {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cos {\relax (x )}}{a} & \text {otherwise} \end {cases} \]

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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