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 = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.58, size = 9, normalized size = 1.00 \[ \frac {\sin \relax (x)}{\cos \relax (x) + 1} \]

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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giac [A]  time = 0.21, size = 4, normalized size = 0.44 \[ \tan \left (\frac {1}{2} \, x\right ) \]

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)

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maple [A]  time = 0.06, size = 5, normalized size = 0.56 \[ \tan \left (\frac {x}{2}\right ) \]

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 = 0.33, size = 9, normalized size = 1.00 \[ \frac {\sin \relax (x)}{\cos \relax (x) + 1} \]

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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mupad [B]  time = 0.54, size = 4, normalized size = 0.44 \[ \mathrm {tan}\left (\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)*(cot(x) + 1/sin(x))),x)

[Out]

tan(x/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc {\relax (x )}}{\cot {\relax (x )} + \csc {\relax (x )}}\, dx \]

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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