3.13 \(\int \frac{\tan (x)}{a+b \cos (x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\log (a+b \cos (x))}{a}-\frac{\log (\cos (x))}{a} \]

[Out]

-(Log[Cos[x]]/a) + Log[a + b*Cos[x]]/a

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Rubi [A]  time = 0.0372535, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2721, 36, 29, 31} \[ \frac{\log (a+b \cos (x))}{a}-\frac{\log (\cos (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/(a + b*Cos[x]),x]

[Out]

-(Log[Cos[x]]/a) + Log[a + b*Cos[x]]/a

Rule 2721

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)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A]  time = 0.0091874, size = 20, normalized size = 1. \[ \frac{\log (a+b \cos (x))}{a}-\frac{\log (\cos (x))}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]/(a + b*Cos[x]),x]

[Out]

-(Log[Cos[x]]/a) + Log[a + b*Cos[x]]/a

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Maple [A]  time = 0.037, size = 21, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( \cos \left ( x \right ) \right ) }{a}}+{\frac{\ln \left ( a+b\cos \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(a+b*cos(x)),x)

[Out]

-ln(cos(x))/a+ln(a+b*cos(x))/a

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Maxima [A]  time = 1.13375, size = 27, normalized size = 1.35 \begin{align*} \frac{\log \left (b \cos \left (x\right ) + a\right )}{a} - \frac{\log \left (\cos \left (x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)),x, algorithm="maxima")

[Out]

log(b*cos(x) + a)/a - log(cos(x))/a

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Fricas [A]  time = 1.53108, size = 53, normalized size = 2.65 \begin{align*} \frac{\log \left (-b \cos \left (x\right ) - a\right ) - \log \left (-\cos \left (x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)),x, algorithm="fricas")

[Out]

(log(-b*cos(x) - a) - log(-cos(x)))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)),x)

[Out]

Integral(tan(x)/(a + b*cos(x)), x)

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Giac [A]  time = 1.33613, size = 30, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{a} - \frac{\log \left ({\left | \cos \left (x\right ) \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(x)/(a+b*cos(x)),x, algorithm="giac")

[Out]

log(abs(b*cos(x) + a))/a - log(abs(cos(x)))/a