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

Optimal. Leaf size=9 \[ \frac{\sin (x)}{\cos (x)+1} \]

[Out]

Sin[x]/(1 + Cos[x])

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Rubi [A]  time = 0.0255203, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3166, 2648} \[ \frac{\sin (x)}{\cos (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(Cot[x] + Csc[x]),x]

[Out]

Sin[x]/(1 + Cos[x])

Rule 3166

Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_)
, x_Symbol] :> Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0]
&& IntegerQ[n]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\csc (x)}{\cot (x)+\csc (x)} \, dx &=\int \frac{1}{1+\cos (x)} \, dx\\ &=\frac{\sin (x)}{1+\cos (x)}\\ \end{align*}

Mathematica [A]  time = 0.0193816, size = 6, normalized size = 0.67 \[ \tan \left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(Cot[x] + Csc[x]),x]

[Out]

Tan[x/2]

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Maple [A]  time = 0.05, size = 5, normalized size = 0.6 \begin{align*} \tan \left ({\frac{x}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(1/2*x)

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Maxima [A]  time = 1.0759, size = 12, normalized size = 1.33 \begin{align*} \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(cos(x) + 1)

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Fricas [A]  time = 0.458424, size = 28, normalized size = 3.11 \begin{align*} \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(cos(x) + 1)

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

[Out]

Integral(csc(x)/(cot(x) + csc(x)), x)

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Giac [A]  time = 1.1713, size = 5, normalized size = 0.56 \begin{align*} \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(1/2*x)