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

Optimal. Leaf size=21 \[ -\frac{1}{3} \log \left (3 \cot \left (\frac{x}{2}+\frac{\pi }{4}\right )+4\right ) \]

[Out]

-Log[4 + 3*Cot[Pi/4 + x/2]]/3

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Rubi [A]  time = 0.0119244, antiderivative size = 21, 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 = {3123, 31} \[ -\frac{1}{3} \log \left (3 \cot \left (\frac{x}{2}+\frac{\pi }{4}\right )+4\right ) \]

Antiderivative was successfully verified.

[In]

Int[(4 + 3*Cos[x] + 4*Sin[x])^(-1),x]

[Out]

-Log[4 + 3*Cot[Pi/4 + x/2]]/3

Rule 3123

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Cot[(d + e*x)/2 + Pi/4], x]}, -Dist[f/e, Subst[Int[1/(a + b*f*x), x], x, Cot[(d + e*x)/2 + Pi/4]/f], x
]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a - c, 0] && NeQ[a - b, 0]

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{1}{4+3 \cos (x)+4 \sin (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{4+3 x} \, dx,x,\cot \left (\frac{\pi }{4}+\frac{x}{2}\right )\right )\\ &=-\frac{1}{3} \log \left (4+3 \cot \left (\frac{\pi }{4}+\frac{x}{2}\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0162603, size = 39, normalized size = 1.86 \[ \frac{1}{3} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\sin \left (\frac{x}{2}\right )+7 \cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Cos[x/2] + Sin[x/2]]/3 - Log[7*Cos[x/2] + Sin[x/2]]/3

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Maple [A]  time = 0.032, size = 20, normalized size = 1. \begin{align*}{\frac{1}{3}\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{3}\ln \left ( \tan \left ({\frac{x}{2}} \right ) +7 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(1+tan(1/2*x))-1/3*ln(tan(1/2*x)+7)

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Maxima [A]  time = 0.932742, size = 39, normalized size = 1.86 \begin{align*} -\frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 7\right ) + \frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(sin(x)/(cos(x) + 1) + 7) + 1/3*log(sin(x)/(cos(x) + 1) + 1)

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Fricas [A]  time = 2.11437, size = 80, normalized size = 3.81 \begin{align*} -\frac{1}{6} \, \log \left (24 \, \cos \left (x\right ) + 7 \, \sin \left (x\right ) + 25\right ) + \frac{1}{6} \, \log \left (\sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*log(24*cos(x) + 7*sin(x) + 25) + 1/6*log(sin(x) + 1)

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Sympy [A]  time = 0.257689, size = 19, normalized size = 0.9 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )}}{3} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + 7 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x/2) + 1)/3 - log(tan(x/2) + 7)/3

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Giac [A]  time = 1.10748, size = 28, normalized size = 1.33 \begin{align*} -\frac{1}{3} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 7 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(abs(tan(1/2*x) + 7)) + 1/3*log(abs(tan(1/2*x) + 1))