3.204 \(\int \frac{\cos (x)}{\cot (x)+\csc (x)} \, dx\)

Optimal. Leaf size=10 \[ \log (\cos (x)+1)-\cos (x) \]

[Out]

-Cos[x] + Log[1 + Cos[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0673322, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4392, 2833, 43} \[ \log (\cos (x)+1)-\cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(Cot[x] + Csc[x]),x]

[Out]

-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 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0067374, size = 20, normalized size = 2. \[ 2 \log \left (\cos \left (\frac{x}{2}\right )\right )-2 \cos ^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(Cot[x] + Csc[x]),x]

[Out]

-2*Cos[x/2]^2 + 2*Log[Cos[x/2]]

________________________________________________________________________________________

Maple [A]  time = 0.082, size = 11, normalized size = 1.1 \begin{align*} -\cos \left ( x \right ) +\ln \left ( \cos \left ( x \right ) +1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)+ln(cos(x)+1)

________________________________________________________________________________________

Maxima [B]  time = 1.60049, size = 46, normalized size = 4.6 \begin{align*} -\frac{2}{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} - \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(cot(x)+csc(x)),x, algorithm="maxima")

[Out]

-2/(sin(x)^2/(cos(x) + 1)^2 + 1) - log(sin(x)^2/(cos(x) + 1)^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 0.485423, size = 45, normalized size = 4.5 \begin{align*} -\cos \left (x\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(cos(x)/(cot(x)+csc(x)),x, algorithm="fricas")

[Out]

-cos(x) + log(1/2*cos(x) + 1/2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\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(cos(x)/(cot(x)+csc(x)),x)

[Out]

Integral(cos(x)/(cot(x) + csc(x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.12215, size = 14, normalized size = 1.4 \begin{align*} -\cos \left (x\right ) + \log \left (\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x) + log(cos(x) + 1)