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(d*x+c)/d/(a+a*cos(d*x+c))

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.91, size = 22, normalized size = 1.00 \[ \frac {\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]

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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giac [A]  time = 0.41, size = 16, normalized size = 0.73 \[ \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a d} \]

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)

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maple [A]  time = 0.04, size = 17, normalized size = 0.77 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.12, size = 23, normalized size = 1.05 \[ \frac {\sin \left (d x + c\right )}{a d {\left (\cos \left (d x + c\right ) + 1\right )}} \]

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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mupad [B]  time = 0.31, size = 16, normalized size = 0.73 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*cos(c + d*x)),x)

[Out]

tan(c/2 + (d*x)/2)/(a*d)

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sympy [A]  time = 0.51, size = 20, normalized size = 0.91 \[ \begin {cases} \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x}{a \cos {\relax (c )} + a} & \text {otherwise} \end {cases} \]

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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