3.283 \(\int \sec (x) \sin (2 x) \, dx\)

Optimal. Leaf size=4 \[ -2 \cos (x) \]

[Out]

-2*Cos[x]

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Rubi [A]  time = 0.0118598, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4287, 2638} \[ -2 \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]*Sin[2*x],x]

[Out]

-2*Cos[x]

Rule 4287

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/e^p, Int[(e*Cos
[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rule 2638

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

Rubi steps

\begin{align*} \int \sec (x) \sin (2 x) \, dx &=2 \int \sin (x) \, dx\\ &=-2 \cos (x)\\ \end{align*}

Mathematica [A]  time = 0.0014547, size = 4, normalized size = 1. \[ -2 \cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]*Sin[2*x],x]

[Out]

-2*Cos[x]

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Maple [A]  time = 0.009, size = 5, normalized size = 1.3 \begin{align*} -2\,\cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/cos(x),x)

[Out]

-2*cos(x)

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Maxima [A]  time = 0.927243, size = 5, normalized size = 1.25 \begin{align*} -2 \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/cos(x),x, algorithm="maxima")

[Out]

-2*cos(x)

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Fricas [A]  time = 1.81245, size = 15, normalized size = 3.75 \begin{align*} -2 \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/cos(x),x, algorithm="fricas")

[Out]

-2*cos(x)

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Sympy [A]  time = 0.707154, size = 5, normalized size = 1.25 \begin{align*} - 2 \cos{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/cos(x),x)

[Out]

-2*cos(x)

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Giac [A]  time = 1.09796, size = 5, normalized size = 1.25 \begin{align*} -2 \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/cos(x),x, algorithm="giac")

[Out]

-2*cos(x)