\(\int \cot (a+b x) \, dx\) [1]

Optimal result

Integrand size = 6, antiderivative size = 11 \[ \int \cot (a+b x) \, dx=\frac {\log (\sin (a+b x))}{b} \]

[Out]

ln(sin(b*x+a))/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3556} \[ \int \cot (a+b x) \, dx=\frac {\log (\sin (a+b x))}{b} \]

[In]

Int[Cot[a + b*x],x]

[Out]

Log[Sin[a + b*x]]/b

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\log (\sin (a+b x))}{b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(11)=22\).

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \cot (a+b x) \, dx=\frac {\log (\cos (a+b x))}{b}+\frac {\log (\tan (a+b x))}{b} \]

[In]

Integrate[Cot[a + b*x],x]

[Out]

Log[Cos[a + b*x]]/b + Log[Tan[a + b*x]]/b

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(-\frac {\ln \left (\cot \left (b x +a \right )^{2}+1\right )}{2 b}\)	\(17\)
default	\(-\frac {\ln \left (\cot \left (b x +a \right )^{2}+1\right )}{2 b}\)	\(17\)
parallelrisch	\(\frac {\ln \left (\tan \left (b x +a \right )\right )+\ln \left (\frac {1}{\sqrt {\sec \left (b x +a \right )^{2}}}\right )}{b}\)	\(24\)
norman	\(\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}-\frac {\ln \left (1+\tan \left (b x +a \right )^{2}\right )}{2 b}\)	\(29\)
risch	\(-i x -\frac {2 i a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\)	\(29\)

[In]

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

[Out]

-1/2/b*ln(cot(b*x+a)^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \cot (a+b x) \, dx=\frac {\log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right )}{2 \, b} \]

[In]

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

[Out]

1/2*log(-1/2*cos(2*b*x + 2*a) + 1/2)/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (8) = 16\).

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.64 \[ \int \cot (a+b x) \, dx=\begin {cases} - \frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} + \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\x \cot {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(b*x+a),x)

[Out]

Piecewise((-log(tan(a + b*x)**2 + 1)/(2*b) + log(tan(a + b*x))/b, Ne(b, 0)), (x*cot(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cot (a+b x) \, dx=\frac {\log \left (\sin \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

log(sin(b*x + a))/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \cot (a+b x) \, dx=\frac {\log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} \]

[In]

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

[Out]

log(abs(sin(b*x + a)))/b

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.18 \[ \int \cot (a+b x) \, dx=-x\,1{}\mathrm {i}+\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b} \]

[In]

int(cot(a + b*x),x)

[Out]

log(exp(a*2i)*exp(b*x*2i) - 1)/b - x*1i