3.3.83 \(\int \sec (x) \sin (2 x) \, dx\) [283]

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

[Out]

-2*cos(x)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, 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 = {4372, 2718} \begin {gather*} -2 \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[x]

Rule 2718

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

Rule 4372

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 4, normalized size = 1.00 \begin {gather*} -2 \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[x]

________________________________________________________________________________________

Mathics [A]
time = 1.90, size = 4, normalized size = 1.00 \begin {gather*} -2 \text {Cos}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[2*x]/Cos[x],x]')

[Out]

-2 Cos[x]

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 5, normalized size = 1.25

method	result	size
derivativedivides	\(-2 \cos \left (x \right )\)	\(5\)
default	\(-2 \cos \left (x \right )\)	\(5\)
risch	\(-2 \cos \left (x \right )\)	\(5\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/cos(x),x,method=_RETURNVERBOSE)

[Out]

-2*cos(x)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 4, normalized size = 1.00 \begin {gather*} -2 \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 4, normalized size = 1.00 \begin {gather*} -2 \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)

________________________________________________________________________________________

Sympy [A]
time = 0.38, size = 5, normalized size = 1.25 \begin {gather*} - 2 \cos {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 5, normalized size = 1.25 \begin {gather*} -2 \cos x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 4, normalized size = 1.00 \begin {gather*} -2\,\cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)

________________________________________________________________________________________