3.2.25 \(\int \cot (a+b x) \, dx\) [125]

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 = {3556} \begin {gather*} \frac {\log (\sin (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \begin {gather*} \frac {\log (\cos (a+b x))+\log (\tan (a+b x))}{b} \end {gather*}

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 = 0.02, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\ln \left (\sin \left (b x +a \right )\right )}{b}\)	\(12\)
default	\(\frac {\ln \left (\sin \left (b x +a \right )\right )}{b}\)	\(12\)
risch	\(-i x -\frac {2 i a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\)	\(29\)
norman	\(\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)/sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sin \left (b x + a\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a),x, algorithm="maxima")

[Out]

log(sin(b*x + a))/b

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Fricas [A]
time = 0.38, size = 13, normalized size = 1.18 \begin {gather*} \frac {\log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a),x, algorithm="fricas")

[Out]

log(1/2*sin(b*x + a))/b

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
time = 0.19, size = 17, normalized size = 1.55 \begin {gather*} \begin {cases} \frac {\log {\left (\sin {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x \cos {\left (a \right )}}{\sin {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a),x)

[Out]

Piecewise((log(sin(a + b*x))/b, Ne(b, 0)), (x*cos(a)/sin(a), True))

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Giac [A]
time = 4.66, size = 12, normalized size = 1.09 \begin {gather*} \frac {\log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+a),x, algorithm="giac")

[Out]

log(abs(sin(b*x + a)))/b

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Mupad [B]
time = 0.41, size = 26, normalized size = 2.36 \begin {gather*} -\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)/sin(a + b*x),x)

[Out]

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

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