\(\int \frac {\cot (x)}{a+a \csc (x)} \, dx\) [5]

Optimal result

Integrand size = 11, antiderivative size = 9 \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\log (1+\sin (x))}{a} \]

[Out]

ln(1+sin(x))/a

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3964, 31} \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\log (\sin (x)+1)}{a} \]

[In]

Int[Cot[x]/(a + a*Csc[x]),x]

[Out]

Log[1 + Sin[x]]/a

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3964

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\sin (x)\right ) \\ & = \frac {\log (1+\sin (x))}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\log (1+\sin (x))}{a} \]

[In]

Integrate[Cot[x]/(a + a*Csc[x]),x]

[Out]

Log[1 + Sin[x]]/a

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.89

method	result	size
derivativedivides	\(-\frac {\ln \left (\csc \left (x \right )\right )-\ln \left (\csc \left (x \right )+1\right )}{a}\)	\(17\)
default	\(-\frac {\ln \left (\csc \left (x \right )\right )-\ln \left (\csc \left (x \right )+1\right )}{a}\)	\(17\)
risch	\(-\frac {i x}{a}+\frac {2 \ln \left (i+{\mathrm e}^{i x}\right )}{a}\)	\(23\)

[In]

int(cot(x)/(a+a*csc(x)),x,method=_RETURNVERBOSE)

[Out]

-1/a*(ln(csc(x))-ln(csc(x)+1))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\log \left (\sin \left (x\right ) + 1\right )}{a} \]

[In]

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

[Out]

log(sin(x) + 1)/a

Sympy [F]

\[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\cot {\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

Integral(cot(x)/(csc(x) + 1), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\log \left (\sin \left (x\right ) + 1\right )}{a} \]

[In]

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

[Out]

log(sin(x) + 1)/a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {\log \left (\sin \left (x\right ) + 1\right )}{a} \]

[In]

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

[Out]

log(sin(x) + 1)/a

Mupad [B] (verification not implemented)

Time = 18.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.78 \[ \int \frac {\cot (x)}{a+a \csc (x)} \, dx=\frac {2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a} \]

[In]

int(cot(x)/(a + a/sin(x)),x)

[Out]

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