3.48 \(\int \frac{1}{a+a \cos (c+d x)} \, dx\)

Optimal. Leaf size=22 \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+a)} \]

[Out]

Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))

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Rubi [A]  time = 0.0123108, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2648} \[ \frac{\sin (c+d x)}{d (a \cos (c+d x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(-1),x]

[Out]

Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0139331, size = 17, normalized size = 0.77 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right )}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^(-1),x]

[Out]

Tan[(c + d*x)/2]/(a*d)

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Maple [A]  time = 0.032, size = 17, normalized size = 0.8 \begin{align*}{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+cos(d*x+c)*a),x)

[Out]

1/d/a*tan(1/2*d*x+1/2*c)

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Maxima [A]  time = 1.20812, size = 31, normalized size = 1.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c)),x, algorithm="maxima")

[Out]

sin(d*x + c)/(a*d*(cos(d*x + c) + 1))

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Fricas [A]  time = 1.56411, size = 53, normalized size = 2.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c)),x, algorithm="fricas")

[Out]

sin(d*x + c)/(a*d*cos(d*x + c) + a*d)

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Sympy [A]  time = 0.597071, size = 20, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x}{a \cos{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c)),x)

[Out]

Piecewise((tan(c/2 + d*x/2)/(a*d), Ne(d, 0)), (x/(a*cos(c) + a), True))

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Giac [A]  time = 1.38707, size = 22, normalized size = 1. \begin{align*} \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c)),x, algorithm="giac")

[Out]

tan(1/2*d*x + 1/2*c)/(a*d)