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

Optimal. Leaf size=16 \[ \frac{\log (\sin (c+d x)+1)}{a d} \]

[Out]

Log[1 + Sin[c + d*x]]/(a*d)

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Rubi [A]  time = 0.0255872, antiderivative size = 16, normalized size of antiderivative = 1., 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 = {2667, 31} \[ \frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Sin[c + d*x]]/(a*d)

Rule 2667

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.0107324, size = 16, normalized size = 1. \[ \frac{\log (\sin (c+d x)+1)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Sin[c + d*x]]/(a*d)

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Maple [A]  time = 0.012, size = 19, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( a+a\sin \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.932045, size = 24, normalized size = 1.5 \begin{align*} \frac{\log \left (a \sin \left (d x + c\right ) + a\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*sin(d*x + c) + a)/(a*d)

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Fricas [A]  time = 1.69334, size = 39, normalized size = 2.44 \begin{align*} \frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(d*x + c) + 1)/(a*d)

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Sympy [A]  time = 0.545384, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} \frac{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(sin(c + d*x) + 1)/(a*d), Ne(d, 0)), (x*cos(c)/(a*sin(c) + a), True))

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Giac [A]  time = 1.33604, size = 26, normalized size = 1.62 \begin{align*} \frac{\log \left ({\left | a \sin \left (d x + c\right ) + a \right |}\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a*sin(d*x + c) + a))/(a*d)