3.34 \(\int \frac{1}{3+5 \cos (c+d x)} \, dx\)

Optimal. Leaf size=65 \[ \frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]

[Out]

-Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/(4*d) + Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/(4*d)

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Rubi [A]  time = 0.0193186, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2659, 206} \[ \frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Cos[c + d*x])^(-1),x]

[Out]

-Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/(4*d) + Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/(4*d)

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{3+5 \cos (c+d x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{8-2 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=-\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}+\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.025905, size = 65, normalized size = 1. \[ \frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (2 \cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*Cos[c + d*x])^(-1),x]

[Out]

-Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/(4*d) + Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/(4*d)

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Maple [A]  time = 0.035, size = 36, normalized size = 0.6 \begin{align*}{\frac{1}{4\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }-{\frac{1}{4\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+5*cos(d*x+c)),x)

[Out]

1/4/d*ln(tan(1/2*d*x+1/2*c)+2)-1/4/d*ln(tan(1/2*d*x+1/2*c)-2)

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Maxima [A]  time = 1.20731, size = 65, normalized size = 1. \begin{align*} \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*cos(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(log(sin(d*x + c)/(cos(d*x + c) + 1) + 2) - log(sin(d*x + c)/(cos(d*x + c) + 1) - 2))/d

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Fricas [A]  time = 1.59046, size = 136, normalized size = 2.09 \begin{align*} \frac{\log \left (\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - \log \left (\frac{3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/8*(log(3/2*cos(d*x + c) + 2*sin(d*x + c) + 5/2) - log(3/2*cos(d*x + c) - 2*sin(d*x + c) + 5/2))/d

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Sympy [A]  time = 0.734198, size = 41, normalized size = 0.63 \begin{align*} \begin{cases} - \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 2 \right )}}{4 d} + \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x}{5 \cos{\left (c \right )} + 3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*cos(d*x+c)),x)

[Out]

Piecewise((-log(tan(c/2 + d*x/2) - 2)/(4*d) + log(tan(c/2 + d*x/2) + 2)/(4*d), Ne(d, 0)), (x/(5*cos(c) + 3), T
rue))

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Giac [A]  time = 1.22231, size = 46, normalized size = 0.71 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \right |}\right ) - \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \right |}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+5*cos(d*x+c)),x, algorithm="giac")

[Out]

1/4*(log(abs(tan(1/2*d*x + 1/2*c) + 2)) - log(abs(tan(1/2*d*x + 1/2*c) - 2)))/d