3.207 \(\int \frac{\sec (x)}{\cot (x)+\csc (x)} \, dx\)

Optimal. Leaf size=11 \[ \log (\cos (x)+1)-\log (\cos (x)) \]

[Out]

-Log[Cos[x]] + Log[1 + Cos[x]]

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Rubi [A]  time = 0.0595973, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4392, 2707, 36, 29, 31} \[ \log (\cos (x)+1)-\log (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]/(Cot[x] + Csc[x]),x]

[Out]

-Log[Cos[x]] + Log[1 + Cos[x]]

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

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

Mathematica [B]  time = 0.0085433, size = 25, normalized size = 2.27 \[ 2 \log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (1-2 \cos ^2\left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]/(Cot[x] + Csc[x]),x]

[Out]

2*Log[Cos[x/2]] - Log[1 - 2*Cos[x/2]^2]

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Maple [A]  time = 0.078, size = 6, normalized size = 0.6 \begin{align*} \ln \left ( 1+\sec \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)/(cot(x)+csc(x)),x)

[Out]

ln(1+sec(x))

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Maxima [B]  time = 1.10753, size = 39, normalized size = 3.55 \begin{align*} -\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \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(sec(x)/(cot(x)+csc(x)),x, algorithm="maxima")

[Out]

-log(sin(x)/(cos(x) + 1) + 1) - log(sin(x)/(cos(x) + 1) - 1)

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Fricas [A]  time = 0.485049, size = 53, normalized size = 4.82 \begin{align*} -\log \left (-\cos \left (x\right )\right ) + \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(cot(x)+csc(x)),x, algorithm="fricas")

[Out]

-log(-cos(x)) + log(1/2*cos(x) + 1/2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (x \right )}}{\cot{\left (x \right )} + \csc{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(cot(x)+csc(x)),x)

[Out]

Integral(sec(x)/(cot(x) + csc(x)), x)

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Giac [A]  time = 1.14099, size = 16, normalized size = 1.45 \begin{align*} \log \left (\cos \left (x\right ) + 1\right ) - \log \left ({\left | \cos \left (x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)/(cot(x)+csc(x)),x, algorithm="giac")

[Out]

log(cos(x) + 1) - log(abs(cos(x)))