3.828 \(\int (-\cos (3 x) \sin (2 x)+\cos (2 x) \sin (3 x)) \, dx\)

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

[Out]

-Cos[x]

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Rubi [A]  time = 0.0173933, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {4284} \[ -\cos (x) \]

Antiderivative was successfully verified.

[In]

Int[-(Cos[3*x]*Sin[2*x]) + Cos[2*x]*Sin[3*x],x]

[Out]

-Cos[x]

Rule 4284

Int[cos[(c_.) + (d_.)*(x_)]*sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[Cos[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Cos[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int (-\cos (3 x) \sin (2 x)+\cos (2 x) \sin (3 x)) \, dx &=-\int \cos (3 x) \sin (2 x) \, dx+\int \cos (2 x) \sin (3 x) \, dx\\ &=-\cos (x)\\ \end{align*}

Mathematica [A]  time = 0.0013774, size = 4, normalized size = 1. \[ -\cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[-(Cos[3*x]*Sin[2*x]) + Cos[2*x]*Sin[3*x],x]

[Out]

-Cos[x]

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Maple [A]  time = 0.05, size = 5, normalized size = 1.3 \begin{align*} -\cos \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(3*x)*sin(2*x)+cos(2*x)*sin(3*x),x)

[Out]

-cos(x)

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Maxima [A]  time = 0.952717, size = 5, normalized size = 1.25 \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(3*x)*sin(2*x)+cos(2*x)*sin(3*x),x, algorithm="maxima")

[Out]

-cos(x)

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Fricas [A]  time = 2.22345, size = 12, normalized size = 3. \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(3*x)*sin(2*x)+cos(2*x)*sin(3*x),x, algorithm="fricas")

[Out]

-cos(x)

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Sympy [B]  time = 0.830159, size = 20, normalized size = 5. \begin{align*} - \sin{\left (2 x \right )} \sin{\left (3 x \right )} - \cos{\left (2 x \right )} \cos{\left (3 x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(3*x)*sin(2*x)+cos(2*x)*sin(3*x),x)

[Out]

-sin(2*x)*sin(3*x) - cos(2*x)*cos(3*x)

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Giac [A]  time = 1.05694, size = 5, normalized size = 1.25 \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(3*x)*sin(2*x)+cos(2*x)*sin(3*x),x, algorithm="giac")

[Out]

-cos(x)