3.9 \(\int \frac{\tan (3 x)}{(1+\cos (3 x))^2} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{3 (\cos (3 x)+1)}-\frac{1}{3} \log (\cos (3 x))+\frac{1}{3} \log (\cos (3 x)+1) \]

[Out]

-1/(3*(1 + Cos[3*x])) - Log[Cos[3*x]]/3 + Log[1 + Cos[3*x]]/3

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Rubi [A]  time = 0.0359734, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2707, 44} \[ -\frac{1}{3 (\cos (3 x)+1)}-\frac{1}{3} \log (\cos (3 x))+\frac{1}{3} \log (\cos (3 x)+1) \]

Antiderivative was successfully verified.

[In]

Int[Tan[3*x]/(1 + Cos[3*x])^2,x]

[Out]

-1/(3*(1 + Cos[3*x])) - Log[Cos[3*x]]/3 + Log[1 + Cos[3*x]]/3

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0709533, size = 49, normalized size = 1.48 \[ \frac{\cos ^4\left (\frac{3 x}{2}\right ) \left (8 \log \left (\cos \left (\frac{3 x}{2}\right )\right )-4 \log (\cos (3 x))\right )-2 \cos ^2\left (\frac{3 x}{2}\right )}{3 (\cos (3 x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[3*x]/(1 + Cos[3*x])^2,x]

[Out]

(-2*Cos[(3*x)/2]^2 + Cos[(3*x)/2]^4*(8*Log[Cos[(3*x)/2]] - 4*Log[Cos[3*x]]))/(3*(1 + Cos[3*x])^2)

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Maple [A]  time = 0.05, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{3+3\,\cos \left ( 3\,x \right ) }}-{\frac{\ln \left ( \cos \left ( 3\,x \right ) \right ) }{3}}+{\frac{\ln \left ( 1+\cos \left ( 3\,x \right ) \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(3*x)/(1+cos(3*x))^2,x)

[Out]

-1/3/(1+cos(3*x))-1/3*ln(cos(3*x))+1/3*ln(1+cos(3*x))

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Maxima [A]  time = 1.12507, size = 36, normalized size = 1.09 \begin{align*} -\frac{1}{3 \,{\left (\cos \left (3 \, x\right ) + 1\right )}} + \frac{1}{3} \, \log \left (\cos \left (3 \, x\right ) + 1\right ) - \frac{1}{3} \, \log \left (\cos \left (3 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(3*x)/(1+cos(3*x))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.41435, size = 132, normalized size = 4. \begin{align*} -\frac{{\left (\cos \left (3 \, x\right ) + 1\right )} \log \left (-\cos \left (3 \, x\right )\right ) -{\left (\cos \left (3 \, x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (3 \, x\right ) + \frac{1}{2}\right ) + 1}{3 \,{\left (\cos \left (3 \, x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(3*x)/(1+cos(3*x))^2,x, algorithm="fricas")

[Out]

-1/3*((cos(3*x) + 1)*log(-cos(3*x)) - (cos(3*x) + 1)*log(1/2*cos(3*x) + 1/2) + 1)/(cos(3*x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (3 x \right )}}{\left (\cos{\left (3 x \right )} + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(3*x)/(1+cos(3*x))**2,x)

[Out]

Integral(tan(3*x)/(cos(3*x) + 1)**2, x)

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Giac [A]  time = 1.46617, size = 53, normalized size = 1.61 \begin{align*} \frac{\cos \left (3 \, x\right ) - 1}{6 \,{\left (\cos \left (3 \, x\right ) + 1\right )}} - \frac{1}{3} \, \log \left ({\left | -\frac{\cos \left (3 \, x\right ) - 1}{\cos \left (3 \, x\right ) + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(3*x)/(1+cos(3*x))^2,x, algorithm="giac")

[Out]

1/6*(cos(3*x) - 1)/(cos(3*x) + 1) - 1/3*log(abs(-(cos(3*x) - 1)/(cos(3*x) + 1) - 1))