3.297 \(\int \frac {\cos (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {\log (a+b \sin (c+d x))}{b d} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2668, 31} \[ \frac {\log (a+b \sin (c+d x))}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]/(a + b*Sin[c + d*x]),x]

[Out]

Log[a + b*Sin[c + d*x]]/(b*d)

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ \frac {\log (a+b \sin (c+d x))}{b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x]),x]

[Out]

Log[a + b*Sin[c + d*x]]/(b*d)

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fricas [A]  time = 0.48, size = 18, normalized size = 1.00 \[ \frac {\log \left (b \sin \left (d x + c\right ) + a\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*sin(d*x + c) + a)/(b*d)

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giac [A]  time = 0.31, size = 19, normalized size = 1.06 \[ \frac {\log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*sin(d*x + c) + a))/(b*d)

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maple [A]  time = 0.00, size = 19, normalized size = 1.06 \[ \frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.78, size = 18, normalized size = 1.00 \[ \frac {\log \left (b \sin \left (d x + c\right ) + a\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*sin(d*x + c) + a)/(b*d)

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mupad [B]  time = 0.06, size = 18, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{b\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*sin(c + d*x))/(b*d)

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sympy [A]  time = 0.62, size = 41, normalized size = 2.28 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \cos {\relax (c )}}{a + b \sin {\relax (c )}} & \text {for}\: d = 0 \\\frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b d} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*cos(c)/a, Eq(b, 0) & Eq(d, 0)), (x*cos(c)/(a + b*sin(c)), Eq(d, 0)), (sin(c + d*x)/(a*d), Eq(b, 0
)), (log(a/b + sin(c + d*x))/(b*d), True))

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