Optimal. Leaf size=46 \[ \frac {5 x}{16}-\frac {1}{12} \sin ^5(2 x) \cos (2 x)-\frac {5}{48} \sin ^3(2 x) \cos (2 x)-\frac {5}{32} \sin (2 x) \cos (2 x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2635, 8} \[ \frac {5 x}{16}-\frac {1}{12} \sin ^5(2 x) \cos (2 x)-\frac {5}{48} \sin ^3(2 x) \cos (2 x)-\frac {5}{32} \sin (2 x) \cos (2 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \sin ^6(2 x) \, dx &=-\frac {1}{12} \cos (2 x) \sin ^5(2 x)+\frac {5}{6} \int \sin ^4(2 x) \, dx\\ &=-\frac {5}{48} \cos (2 x) \sin ^3(2 x)-\frac {1}{12} \cos (2 x) \sin ^5(2 x)+\frac {5}{8} \int \sin ^2(2 x) \, dx\\ &=-\frac {5}{32} \cos (2 x) \sin (2 x)-\frac {5}{48} \cos (2 x) \sin ^3(2 x)-\frac {1}{12} \cos (2 x) \sin ^5(2 x)+\frac {5 \int 1 \, dx}{16}\\ &=\frac {5 x}{16}-\frac {5}{32} \cos (2 x) \sin (2 x)-\frac {5}{48} \cos (2 x) \sin ^3(2 x)-\frac {1}{12} \cos (2 x) \sin ^5(2 x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 30, normalized size = 0.65 \[ \frac {5 x}{16}-\frac {15}{128} \sin (4 x)+\frac {3}{128} \sin (8 x)-\frac {1}{384} \sin (12 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 33, normalized size = 0.72 \[ -\frac {1}{96} \, {\left (8 \, \cos \left (2 \, x\right )^{5} - 26 \, \cos \left (2 \, x\right )^{3} + 33 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, x\right ) + \frac {5}{16} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.02, size = 22, normalized size = 0.48 \[ \frac {5}{16} \, x - \frac {1}{384} \, \sin \left (12 \, x\right ) + \frac {3}{128} \, \sin \left (8 \, x\right ) - \frac {15}{128} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 32, normalized size = 0.70 \[ \frac {5 x}{16}-\frac {\left (\sin ^{5}\left (2 x \right )+\frac {5 \left (\sin ^{3}\left (2 x \right )\right )}{4}+\frac {15 \sin \left (2 x \right )}{8}\right ) \cos \left (2 x \right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 24, normalized size = 0.52 \[ \frac {1}{96} \, \sin \left (4 \, x\right )^{3} + \frac {5}{16} \, x + \frac {3}{128} \, \sin \left (8 \, x\right ) - \frac {1}{8} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.21, size = 22, normalized size = 0.48 \[ \frac {5\,x}{16}-\frac {15\,\sin \left (4\,x\right )}{128}+\frac {3\,\sin \left (8\,x\right )}{128}-\frac {\sin \left (12\,x\right )}{384} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.07, size = 46, normalized size = 1.00 \[ \frac {5 x}{16} - \frac {\sin ^{5}{\left (2 x \right )} \cos {\left (2 x \right )}}{12} - \frac {5 \sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{48} - \frac {5 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________