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

Optimal. Leaf size=11 \[ \frac {\log (a+b \sin (x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 11, 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 = {2668, 31} \[ \frac {\log (a+b \sin (x))}{b} \]

Antiderivative was successfully verified.

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

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\log (a+b \sin (x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(a + b*Sin[x]),x]

[Out]

Log[a + b*Sin[x]]/b

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fricas [A]  time = 1.97, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \sin \relax (x) + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*sin(x) + a)/b

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giac [A]  time = 0.14, size = 12, normalized size = 1.09 \[ \frac {\log \left ({\left | b \sin \relax (x) + a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*sin(x) + a))/b

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maple [A]  time = 0.03, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \sin \relax (x )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \sin \relax (x) + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*sin(x) + a)/b

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mupad [B]  time = 0.03, size = 11, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\sin \relax (x)\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(a + b*sin(x)),x)

[Out]

log(a + b*sin(x))/b

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sympy [A]  time = 0.32, size = 14, normalized size = 1.27 \[ \begin {cases} \frac {\log {\left (\frac {a}{b} + \sin {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\sin {\relax (x )}}{a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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