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

Optimal. Leaf size=15 \[ \frac{1}{4} \log \left (4 \tan \left (\frac{x}{2}\right )+3\right ) \]

[Out]

Log[3 + 4*Tan[x/2]]/4

________________________________________________________________________________________

Rubi [A]  time = 0.0131235, antiderivative size = 15, 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 = {3124, 31} \[ \frac{1}{4} \log \left (4 \tan \left (\frac{x}{2}\right )+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 3*Cos[x] + 4*Sin[x])^(-1),x]

[Out]

Log[3 + 4*Tan[x/2]]/4

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 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}{3+3 \cos (x)+4 \sin (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{6+8 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{4} \log \left (3+4 \tan \left (\frac{x}{2}\right )\right )\\ \end{align*}

Mathematica [B]  time = 0.017616, size = 34, normalized size = 2.27 \[ \frac{1}{4} \log \left (4 \sin \left (\frac{x}{2}\right )+3 \cos \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[Cos[x/2]]/4 + Log[3*Cos[x/2] + 4*Sin[x/2]]/4

________________________________________________________________________________________

Maple [A]  time = 0.029, size = 12, normalized size = 0.8 \begin{align*}{\frac{1}{4}\ln \left ( 3+4\,\tan \left ( x/2 \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(3+4*tan(1/2*x))

________________________________________________________________________________________

Maxima [A]  time = 0.999468, size = 20, normalized size = 1.33 \begin{align*} \frac{1}{4} \, \log \left (\frac{4 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(4*sin(x)/(cos(x) + 1) + 3)

________________________________________________________________________________________

Fricas [B]  time = 2.06574, size = 95, normalized size = 6.33 \begin{align*} -\frac{1}{8} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (-\frac{7}{2} \, \cos \left (x\right ) + 12 \, \sin \left (x\right ) + \frac{25}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*log(1/2*cos(x) + 1/2) + 1/8*log(-7/2*cos(x) + 12*sin(x) + 25/2)

________________________________________________________________________________________

Sympy [A]  time = 0.250092, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + \frac{3}{4} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x/2) + 3/4)/4

________________________________________________________________________________________

Giac [A]  time = 1.10376, size = 16, normalized size = 1.07 \begin{align*} \frac{1}{4} \, \log \left ({\left | 4 \, \tan \left (\frac{1}{2} \, x\right ) + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(4*tan(1/2*x) + 3))