\(\int \frac {\cos (x)}{a+b \sin (x)} \, dx\) [665]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\log (a+b \sin (x))}{b} \]

[Out]

ln(a+b*sin(x))/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2747, 31} \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\log (a+b \sin (x))}{b} \]

[In]

Int[Cos[x]/(a + b*Sin[x]),x]

[Out]

Log[a + b*Sin[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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (x)\right )}{b} \\ & = \frac {\log (a+b \sin (x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\log (a+b \sin (x))}{b} \]

[In]

Integrate[Cos[x]/(a + b*Sin[x]),x]

[Out]

Log[a + b*Sin[x]]/b

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\ln \left (a +b \sin \left (x \right )\right )}{b}\)	\(12\)
default	\(\frac {\ln \left (a +b \sin \left (x \right )\right )}{b}\)	\(12\)
parallelrisch	\(\frac {-\ln \left (\frac {1}{\cos \left (x \right )+1}\right )+\ln \left (\frac {a +b \sin \left (x \right )}{\cos \left (x \right )+1}\right )}{b}\)	\(29\)
risch	\(-\frac {i x}{b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{b}-1\right )}{b}\)	\(33\)
norman	\(\frac {\ln \left (\tan \left (\frac {x}{2}\right )^{2} a +2 b \tan \left (\frac {x}{2}\right )+a \right )}{b}-\frac {\ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{b}\)	\(38\)

[In]

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

[Out]

ln(a+b*sin(x))/b

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\log \left (b \sin \left (x\right ) + a\right )}{b} \]

[In]

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

[Out]

log(b*sin(x) + a)/b

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\begin {cases} \frac {\log {\left (\frac {a}{b} + \sin {\left (x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\sin {\left (x \right )}}{a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\log \left (b \sin \left (x\right ) + a\right )}{b} \]

[In]

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

[Out]

log(b*sin(x) + a)/b

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\log \left ({\left | b \sin \left (x\right ) + a \right |}\right )}{b} \]

[In]

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

[Out]

log(abs(b*sin(x) + a))/b

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a+b \sin (x)} \, dx=\frac {\ln \left (a+b\,\sin \left (x\right )\right )}{b} \]

[In]

int(cos(x)/(a + b*sin(x)),x)

[Out]

log(a + b*sin(x))/b