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 = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.02, 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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fricas [A]  time = 0.50, size = 22, normalized size = 1.83 \[ -\frac {\cos \relax (x) - \sin \relax (x) + 1}{a \cos \relax (x) + a \sin \relax (x) + a} \]

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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giac [A]  time = 0.45, size = 13, normalized size = 1.08 \[ -\frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]

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))

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maple [A]  time = 0.20, size = 14, normalized size = 1.17 \[ -\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]

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 = 0.34, size = 16, normalized size = 1.33 \[ -\frac {2}{a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1}} \]

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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mupad [B]  time = 0.17, size = 13, normalized size = 1.08 \[ -\frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)*(a + a/sin(x))),x)

[Out]

-2/(a*(tan(x/2) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc {\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]

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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