3.298 \(\int \frac{1}{\cot (x)+\csc (x)} \, dx\)

Optimal. Leaf size=7 \[ -\log (\cos (x)+1) \]

[Out]

-Log[1 + Cos[x]]

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Rubi [A]  time = 0.0269082, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3160, 2667, 31} \[ -\log (\cos (x)+1) \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x] + Csc[x])^(-1),x]

[Out]

-Log[1 + Cos[x]]

Rule 3160

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(-1), x_Symbol] :> Int[Sin[d + e*x
]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(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]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

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

Mathematica [A]  time = 0.0119092, size = 9, normalized size = 1.29 \[ -2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x] + Csc[x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.047, size = 8, normalized size = 1.1 \begin{align*} -\ln \left ( 1+\cos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)+csc(x)),x)

[Out]

-ln(1+cos(x))

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Maxima [B]  time = 1.45563, size = 19, normalized size = 2.71 \begin{align*} \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(1/(cot(x)+csc(x)),x, algorithm="maxima")

[Out]

log(sin(x)^2/(cos(x) + 1)^2 + 1)

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Fricas [A]  time = 2.13351, size = 32, normalized size = 4.57 \begin{align*} -\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(1/(cot(x)+csc(x)),x, algorithm="fricas")

[Out]

-log(1/2*cos(x) + 1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x)),x)

[Out]

Integral(1/(cot(x) + csc(x)), x)

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Giac [A]  time = 1.1338, size = 9, normalized size = 1.29 \begin{align*} -\log \left (\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)+csc(x)),x, algorithm="giac")

[Out]

-log(cos(x) + 1)