3.30 \(\int \cos (3 x) \cos (4 x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x) \]

[Out]

Sin[x]/2 + Sin[7*x]/14

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Rubi [A]  time = 0.0082189, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4283} \[ \frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]*Cos[4*x],x]

[Out]

Sin[x]/2 + Sin[7*x]/14

Rule 4283

Int[cos[(a_.) + (b_.)*(x_)]*cos[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
+ Simp[Sin[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) \cos (4 x) \, dx &=\frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x)\\ \end{align*}

Mathematica [A]  time = 0.0060621, size = 15, normalized size = 1. \[ \frac{\sin (x)}{2}+\frac{1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]*Cos[4*x],x]

[Out]

Sin[x]/2 + Sin[7*x]/14

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Maple [A]  time = 0., size = 12, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( x \right ) }{2}}+{\frac{\sin \left ( 7\,x \right ) }{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*cos(4*x),x)

[Out]

1/2*sin(x)+1/14*sin(7*x)

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Maxima [A]  time = 0.939458, size = 15, normalized size = 1. \begin{align*} \frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*cos(4*x),x, algorithm="maxima")

[Out]

1/14*sin(7*x) + 1/2*sin(x)

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Fricas [B]  time = 2.12828, size = 78, normalized size = 5.2 \begin{align*} \frac{1}{7} \,{\left (32 \, \cos \left (x\right )^{6} - 40 \, \cos \left (x\right )^{4} + 12 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*cos(4*x),x, algorithm="fricas")

[Out]

1/7*(32*cos(x)^6 - 40*cos(x)^4 + 12*cos(x)^2 + 3)*sin(x)

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Sympy [B]  time = 0.52267, size = 26, normalized size = 1.73 \begin{align*} - \frac{3 \sin{\left (3 x \right )} \cos{\left (4 x \right )}}{7} + \frac{4 \sin{\left (4 x \right )} \cos{\left (3 x \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*cos(4*x),x)

[Out]

-3*sin(3*x)*cos(4*x)/7 + 4*sin(4*x)*cos(3*x)/7

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Giac [A]  time = 1.06341, size = 15, normalized size = 1. \begin{align*} \frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*cos(4*x),x, algorithm="giac")

[Out]

1/14*sin(7*x) + 1/2*sin(x)