\(\int \cos (3 x) \cos (4 x) \, dx\) [108]

Optimal result

Integrand size = 9, antiderivative size = 15 \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{14} \sin (7 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4368} \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{14} \sin (7 x) \]

[In]

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

[Out]

Sin[x]/2 + Sin[7*x]/14

Rule 4368

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*} \text {integral}& = \frac {\sin (x)}{2}+\frac {1}{14} \sin (7 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {\sin (x)}{2}+\frac {1}{14} \sin (7 x) \]

[In]

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

[Out]

Sin[x]/2 + Sin[7*x]/14

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (7 x \right )}{14}\)	\(12\)
risch	\(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (7 x \right )}{14}\)	\(12\)
parallelrisch	\(\frac {\sin \left (x \right )}{2}+\frac {\sin \left (7 x \right )}{14}\)	\(12\)
norman	\(\frac {-\frac {8 \tan \left (2 x \right ) \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{2}\left (2 x \right )\right ) \tan \left (\frac {3 x}{2}\right )}{7}+\frac {8 \tan \left (2 x \right )}{7}-\frac {6 \tan \left (\frac {3 x}{2}\right )}{7}}{\left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right ) \left (1+\tan ^{2}\left (2 x \right )\right )}\)	\(59\)

[In]

int(cos(3*x)*cos(4*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {1}{7} \, {\left (32 \, \cos \left (x\right )^{6} - 40 \, \cos \left (x\right )^{4} + 12 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right ) \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \cos (3 x) \cos (4 x) \, dx=- \frac {3 \sin {\left (3 x \right )} \cos {\left (4 x \right )}}{7} + \frac {4 \sin {\left (4 x \right )} \cos {\left (3 x \right )}}{7} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {1}{14} \, \sin \left (7 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]

[In]

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

[Out]

1/14*sin(7*x) + 1/2*sin(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {1}{14} \, \sin \left (7 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]

[In]

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

[Out]

1/14*sin(7*x) + 1/2*sin(x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \cos (3 x) \cos (4 x) \, dx=\frac {\sin \left (7\,x\right )}{14}+\frac {\sin \left (x\right )}{2} \]

[In]

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

[Out]

sin(7*x)/14 + sin(x)/2