3.5 \(\int \frac{\csc (x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\cot (x)}{a \csc (x)+a} \]

[Out]

-(Cot[x]/(a + a*Csc[x]))

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Rubi [A]  time = 0.0198414, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3794} \[ -\frac{\cot (x)}{a \csc (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(a + a*Csc[x]),x]

[Out]

-(Cot[x]/(a + a*Csc[x]))

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\csc (x)}{a+a \csc (x)} \, dx &=-\frac{\cot (x)}{a+a \csc (x)}\\ \end{align*}

Mathematica [B]  time = 0.0239544, size = 26, normalized size = 2.17 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{a \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(a + a*Csc[x]),x]

[Out]

(2*Sin[x/2])/(a*(Cos[x/2] + Sin[x/2]))

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Maple [A]  time = 0.024, size = 14, normalized size = 1.2 \begin{align*} -2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)/(a+a*csc(x)),x)

[Out]

-2/a/(tan(1/2*x)+1)

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Maxima [A]  time = 1.01956, size = 22, normalized size = 1.83 \begin{align*} -\frac{2}{a + \frac{a \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)/(a+a*csc(x)),x, algorithm="maxima")

[Out]

-2/(a + a*sin(x)/(cos(x) + 1))

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Fricas [A]  time = 0.444167, size = 68, normalized size = 5.67 \begin{align*} -\frac{\cos \left (x\right ) - \sin \left (x\right ) + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)/(a+a*csc(x)),x)

[Out]

Integral(csc(x)/(csc(x) + 1), x)/a

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Giac [A]  time = 1.3368, size = 18, normalized size = 1.5 \begin{align*} -\frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*(tan(1/2*x) + 1))