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

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 = 0.0044883, 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 = 0.006467, 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 = 0.01, size = 12, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sec \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)*sin(b*x+a),x)

[Out]

1/b*ln(sec(b*x+a))

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Maxima [A]  time = 0.988365, size = 24, normalized size = 2. \begin{align*} -\frac{\log \left (-\sin \left (b x + a\right )^{2} + 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-sin(b*x + a)^2 + 1)/b

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Fricas [A]  time = 1.92592, size = 31, normalized size = 2.58 \begin{align*} -\frac{\log \left (-\cos \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-cos(b*x + a))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b x \right )} \sec{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)*sin(b*x+a),x)

[Out]

Integral(sin(a + b*x)*sec(a + b*x), x)

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Giac [A]  time = 1.20248, size = 24, normalized size = 2. \begin{align*} -\frac{\log \left (\frac{{\left | \cos \left (b x + a\right ) \right |}}{{\left | b \right |}}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(cos(b*x + a))/abs(b))/b