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

Optimal. Leaf size=11 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]

[Out]

ArcTanh[Sin[a + b*x]]/b

________________________________________________________________________________________

Rubi [A]  time = 0.0042149, 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 = {3770} \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sec (a+b x) \, dx &=\frac{\tanh ^{-1}(\sin (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0022212, size = 11, normalized size = 1. \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x],x]

[Out]

ArcTanh[Sin[a + b*x]]/b

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 19, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10381, size = 24, normalized size = 2.18 \begin{align*} \frac{\log \left (\sec \left (b x + a\right ) + \tan \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),x, algorithm="maxima")

[Out]

log(sec(b*x + a) + tan(b*x + a))/b

________________________________________________________________________________________

Fricas [B]  time = 1.52565, size = 76, normalized size = 6.91 \begin{align*} \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 5.91471, size = 36, normalized size = 3.27 \begin{align*} \begin{cases} \frac{\log{\left (\tan{\left (a + b x \right )} + \sec{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x \left (\tan{\left (a \right )} \sec{\left (a \right )} + \sec ^{2}{\left (a \right )}\right )}{\tan{\left (a \right )} + \sec{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(tan(a + b*x) + sec(a + b*x))/b, Ne(b, 0)), (x*(tan(a)*sec(a) + sec(a)**2)/(tan(a) + sec(a)), Tr
ue))

________________________________________________________________________________________

Giac [B]  time = 1.31224, size = 59, normalized size = 5.36 \begin{align*} \frac{\log \left ({\left | \frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) + 2 \right |}\right ) - \log \left ({\left | \frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) - 2 \right |}\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(abs(1/sin(b*x + a) + sin(b*x + a) + 2)) - log(abs(1/sin(b*x + a) + sin(b*x + a) - 2)))/b