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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.45, size = 27, normalized size = 2.45 \[ \frac {\log \left (\frac {\tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(b*x+a),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.34, size = 56, normalized size = 5.09 \[ \frac {\log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(b*x+a),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.01, size = 29, normalized size = 2.64 \[ -\frac {\ln \left (\tan ^{2}\left (b x +a \right )+1\right )}{2 b}+\frac {\ln \left (\tan \left (b x +a \right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(b*x+a),x)

[Out]

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

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maxima [A]  time = 0.42, size = 11, normalized size = 1.00 \[ \frac {\log \left (\sin \left (b x + a\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(b*x+a),x, algorithm="maxima")

[Out]

log(sin(b*x + a))/b

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mupad [B]  time = 0.19, size = 28, normalized size = 2.55 \[ \frac {\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b}-\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.33, size = 29, normalized size = 2.64 \[ \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} + \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\tan {\relax (a )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(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/tan(a), True))

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