3.46 \(\int \sec (a+b x) \tan (a+b x) \, dx\)

Optimal. Leaf size=10 \[ \frac{\sec (a+b x)}{b} \]

[Out]

Sec[a + b*x]/b

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Rubi [A]  time = 0.0107023, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2606, 8} \[ \frac{\sec (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]*Tan[a + b*x],x]

[Out]

Sec[a + b*x]/b

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sec (a+b x) \tan (a+b x) \, dx &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\sec (a+b x))}{b}\\ &=\frac{\sec (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.006631, size = 10, normalized size = 1. \[ \frac{\sec (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]*Tan[a + b*x],x]

[Out]

Sec[a + b*x]/b

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Maple [A]  time = 0.009, size = 11, normalized size = 1.1 \begin{align*}{\frac{\sec \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^2*sin(b*x+a),x)

[Out]

sec(b*x+a)/b

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Maxima [A]  time = 0.974452, size = 16, normalized size = 1.6 \begin{align*} \frac{1}{b \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^2*sin(b*x+a),x, algorithm="maxima")

[Out]

1/(b*cos(b*x + a))

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Fricas [A]  time = 1.87666, size = 27, normalized size = 2.7 \begin{align*} \frac{1}{b \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^2*sin(b*x+a),x, algorithm="fricas")

[Out]

1/(b*cos(b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**2*sin(b*x+a),x)

[Out]

Integral(sin(a + b*x)*sec(a + b*x)**2, x)

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Giac [A]  time = 1.1312, size = 16, normalized size = 1.6 \begin{align*} \frac{1}{b \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^2*sin(b*x+a),x, algorithm="giac")

[Out]

1/(b*cos(b*x + a))