Optimal. Leaf size=73 \[ \frac {32 \cos \left (\frac {2 x}{3}\right )}{5 \sqrt {1-\sin \left (\frac {2 x}{3}\right )}}+\frac {8}{5} \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}+\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2726, 2725}
\begin {gather*} \frac {3}{5} \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2} \cos \left (\frac {2 x}{3}\right )+\frac {8}{5} \sqrt {1-\sin \left (\frac {2 x}{3}\right )} \cos \left (\frac {2 x}{3}\right )+\frac {32 \cos \left (\frac {2 x}{3}\right )}{5 \sqrt {1-\sin \left (\frac {2 x}{3}\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2725
Rule 2726
Rubi steps
\begin {align*} \int \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{5/2} \, dx &=\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2}+\frac {8}{5} \int \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2} \, dx\\ &=\frac {8}{5} \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}+\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2}+\frac {32}{15} \int \sqrt {1-\sin \left (\frac {2 x}{3}\right )} \, dx\\ &=\frac {32 \cos \left (\frac {2 x}{3}\right )}{5 \sqrt {1-\sin \left (\frac {2 x}{3}\right )}}+\frac {8}{5} \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}+\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 76, normalized size = 1.04 \begin {gather*} \frac {\left (1-\sin \left (\frac {2 x}{3}\right )\right )^{5/2} \left (150 \cos \left (\frac {x}{3}\right )+25 \cos (x)-3 \cos \left (\frac {5 x}{3}\right )+150 \sin \left (\frac {x}{3}\right )-25 \sin (x)-3 \sin \left (\frac {5 x}{3}\right )\right )}{20 \left (\cos \left (\frac {x}{3}\right )-\sin \left (\frac {x}{3}\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 47, normalized size = 0.64
method | result | size |
default | \(-\frac {\left (-1+\sin \left (\frac {2 x}{3}\right )\right ) \left (\sin \left (\frac {2 x}{3}\right )+1\right ) \left (3 \left (\sin ^{2}\left (\frac {2 x}{3}\right )\right )-14 \sin \left (\frac {2 x}{3}\right )+43\right )}{5 \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.10, size = 71, normalized size = 0.97 \begin {gather*} -\frac {{\left (3 \, \cos \left (\frac {2}{3} \, x\right )^{3} - 11 \, \cos \left (\frac {2}{3} \, x\right )^{2} + {\left (3 \, \cos \left (\frac {2}{3} \, x\right )^{2} + 14 \, \cos \left (\frac {2}{3} \, x\right ) - 32\right )} \sin \left (\frac {2}{3} \, x\right ) - 46 \, \cos \left (\frac {2}{3} \, x\right ) - 32\right )} \sqrt {-\sin \left (\frac {2}{3} \, x\right ) + 1}}{5 \, {\left (\cos \left (\frac {2}{3} \, x\right ) - \sin \left (\frac {2}{3} \, x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (1 - \sin {\left (\frac {2 x}{3} \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.40, size = 72, normalized size = 0.99 \begin {gather*} -\frac {1}{20} \, \sqrt {2} {\left (150 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) - 25 \, \cos \left (-\frac {3}{4} \, \pi + x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) + 3 \, \cos \left (-\frac {5}{4} \, \pi + \frac {5}{3} \, x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) - 128 \, \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-\sin \left (\frac {2\,x}{3}\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________