Optimal. Leaf size=30 \[ \frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0306922, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4355, 2637} \[ \frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4355
Rule 2637
Rubi steps
\begin{align*} \int \cos (x) \cos (2 x) \cos (3 x) \, dx &=\int \left (\frac{1}{4}+\frac{1}{4} \cos (2 x)+\frac{1}{4} \cos (4 x)+\frac{1}{4} \cos (6 x)\right ) \, dx\\ &=\frac{x}{4}+\frac{1}{4} \int \cos (2 x) \, dx+\frac{1}{4} \int \cos (4 x) \, dx+\frac{1}{4} \int \cos (6 x) \, dx\\ &=\frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x)\\ \end{align*}
Mathematica [A] time = 0.0084162, size = 30, normalized size = 1. \[ \frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 23, normalized size = 0.8 \begin{align*}{\frac{x}{4}}+{\frac{\sin \left ( 2\,x \right ) }{8}}+{\frac{\sin \left ( 4\,x \right ) }{16}}+{\frac{\sin \left ( 6\,x \right ) }{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.931044, size = 30, normalized size = 1. \begin{align*} \frac{1}{4} \, x + \frac{1}{24} \, \sin \left (6 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97036, size = 81, normalized size = 2.7 \begin{align*} \frac{1}{12} \,{\left (16 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{4} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 14.4865, size = 112, normalized size = 3.73 \begin{align*} - \frac{x \sin{\left (x \right )} \sin{\left (2 x \right )} \cos{\left (3 x \right )}}{4} + \frac{x \sin{\left (x \right )} \sin{\left (3 x \right )} \cos{\left (2 x \right )}}{4} + \frac{x \sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (x \right )}}{4} + \frac{x \cos{\left (x \right )} \cos{\left (2 x \right )} \cos{\left (3 x \right )}}{4} + \frac{\sin{\left (x \right )} \sin{\left (2 x \right )} \sin{\left (3 x \right )}}{24} - \frac{\sin{\left (2 x \right )} \cos{\left (x \right )} \cos{\left (3 x \right )}}{8} + \frac{\sin{\left (3 x \right )} \cos{\left (x \right )} \cos{\left (2 x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.06672, size = 30, normalized size = 1. \begin{align*} \frac{1}{4} \, x + \frac{1}{24} \, \sin \left (6 \, x\right ) + \frac{1}{16} \, \sin \left (4 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]