Optimal. Leaf size=19 \[ \sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713}
\begin {gather*} \frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2713
Rubi steps
\begin {align*} \int \cos ^5(x) \, dx &=-\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )\\ &=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 23, normalized size = 1.21 \begin {gather*} \frac {5 \sin (x)}{8}+\frac {5}{48} \sin (3 x)+\frac {1}{80} \sin (5 x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 17, normalized size = 0.89
method | result | size |
default | \(\frac {\left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{5}\) | \(17\) |
risch | \(\frac {5 \sin \left (x \right )}{8}+\frac {\sin \left (5 x \right )}{80}+\frac {5 \sin \left (3 x \right )}{48}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.68, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, \sin \left (x\right )^{5} - \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.74, size = 18, normalized size = 0.95 \begin {gather*} \frac {1}{15} \, {\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 17, normalized size = 0.89 \begin {gather*} \frac {\sin ^{5}{\left (x \right )}}{5} - \frac {2 \sin ^{3}{\left (x \right )}}{3} + \sin {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.82, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, \sin \left (x\right )^{5} - \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.00, size = 21, normalized size = 1.11 \begin {gather*} \frac {\sin \left (x\right )\,{\cos \left (x\right )}^4}{5}+\frac {4\,\sin \left (x\right )\,{\cos \left (x\right )}^2}{15}+\frac {8\,\sin \left (x\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________