3.1 \(\int \cot (a+b x) \, dx\)

Optimal. Leaf size=11 \[ \frac{\log (\sin (a+b x))}{b} \]

[Out]

Log[Sin[a + b*x]]/b

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Rubi [A]  time = 0.004614, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3475} \[ \frac{\log (\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x],x]

[Out]

Log[Sin[a + b*x]]/b

Rule 3475

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*} \int \cot (a+b x) \, dx &=\frac{\log (\sin (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0147557, size = 19, normalized size = 1.73 \[ \frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.013, size = 17, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( \left ( \cot \left ( bx+a \right ) \right ) ^{2}+1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0295, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (\sin \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sin(b*x + a))/b

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Fricas [A]  time = 1.91427, size = 54, normalized size = 4.91 \begin{align*} \frac{\log \left (-\frac{1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac{1}{2}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.190214, size = 29, normalized size = 2.64 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.09626, size = 16, normalized size = 1.45 \begin{align*} \frac{\log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(sin(b*x + a)))/b