3.6 \(\int \frac{1}{5+3 \cos (x)+4 \sin (x)} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{\tan \left (\frac{x}{2}\right )+2} \]

[Out]

-(2 + Tan[x/2])^(-1)

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Rubi [A]  time = 0.0113994, antiderivative size = 21, normalized size of antiderivative = 1.75, 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 = {3114} \[ -\frac{4-5 \sin (x)}{4 (4 \cos (x)-3 \sin (x))} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*Cos[x] + 4*Sin[x])^(-1),x]

[Out]

-(4 - 5*Sin[x])/(4*(4*Cos[x] - 3*Sin[x]))

Rule 3114

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

Rubi steps

\begin{align*} \int \frac{1}{5+3 \cos (x)+4 \sin (x)} \, dx &=-\frac{4-5 \sin (x)}{4 (4 \cos (x)-3 \sin (x))}\\ \end{align*}

Mathematica [B]  time = 0.0312558, size = 26, normalized size = 2.17 \[ \frac{\sin \left (\frac{x}{2}\right )}{2 \sin \left (\frac{x}{2}\right )+4 \cos \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*Cos[x] + 4*Sin[x])^(-1),x]

[Out]

Sin[x/2]/(4*Cos[x/2] + 2*Sin[x/2])

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Maple [A]  time = 0.03, size = 11, normalized size = 0.9 \begin{align*} - \left ( 2+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*cos(x)+4*sin(x)),x)

[Out]

-1/(2+tan(1/2*x))

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Maxima [A]  time = 0.932105, size = 20, normalized size = 1.67 \begin{align*} -\frac{1}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cos(x)+4*sin(x)),x, algorithm="maxima")

[Out]

-1/(sin(x)/(cos(x) + 1) + 2)

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Fricas [A]  time = 1.74958, size = 73, normalized size = 6.08 \begin{align*} -\frac{\cos \left (x\right ) - 2 \, \sin \left (x\right ) + 1}{5 \,{\left (2 \, \cos \left (x\right ) + \sin \left (x\right ) + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cos(x)+4*sin(x)),x, algorithm="fricas")

[Out]

-1/5*(cos(x) - 2*sin(x) + 1)/(2*cos(x) + sin(x) + 2)

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Sympy [A]  time = 0.442646, size = 8, normalized size = 0.67 \begin{align*} - \frac{1}{\tan{\left (\frac{x}{2} \right )} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cos(x)+4*sin(x)),x)

[Out]

-1/(tan(x/2) + 2)

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Giac [A]  time = 1.09093, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{\tan \left (\frac{1}{2} \, x\right ) + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*cos(x)+4*sin(x)),x, algorithm="giac")

[Out]

-1/(tan(1/2*x) + 2)