3.109 \(\int \tan (a+b x) \, dx\)

Optimal. Leaf size=12 \[ -\frac{\log (\cos (a+b x))}{b} \]

[Out]

-(Log[Cos[a + b*x]]/b)

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Rubi [A]  time = 0.0040621, antiderivative size = 12, 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} \[ -\frac{\log (\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Tan[a + b*x],x]

[Out]

-(Log[Cos[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 \tan (a+b x) \, dx &=-\frac{\log (\cos (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0088248, size = 12, normalized size = 1. \[ -\frac{\log (\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + b*x],x]

[Out]

-(Log[Cos[a + b*x]]/b)

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Maple [A]  time = 0.002, size = 17, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( bx+a \right ) \right ) ^{2} \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.936232, size = 15, normalized size = 1.25 \begin{align*} \frac{\log \left (\sec \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sec(b*x + a))/b

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Fricas [A]  time = 1.80857, size = 49, normalized size = 4.08 \begin{align*} -\frac{\log \left (\frac{1}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(1/(tan(b*x + a)^2 + 1))/b

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Sympy [A]  time = 0.123411, size = 19, normalized size = 1.58 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} & \text{for}\: b \neq 0 \\x \tan{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07097, size = 18, normalized size = 1.5 \begin{align*} -\frac{\log \left ({\left | \cos \left (b x + a\right ) \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(cos(b*x + a)))/b