Optimal. Leaf size=46 \[ \frac {5 a^3 x}{16}+\frac {1}{6} a^3 \sin (x) \cos ^5(x)+\frac {5}{24} a^3 \sin (x) \cos ^3(x)+\frac {5}{16} a^3 \sin (x) \cos (x) \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3175, 2635, 8} \[ \frac {5 a^3 x}{16}+\frac {1}{6} a^3 \sin (x) \cos ^5(x)+\frac {5}{24} a^3 \sin (x) \cos ^3(x)+\frac {5}{16} a^3 \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3175
Rubi steps
\begin {align*} \int \left (a-a \sin ^2(x)\right )^3 \, dx &=a^3 \int \cos ^6(x) \, dx\\ &=\frac {1}{6} a^3 \cos ^5(x) \sin (x)+\frac {1}{6} \left (5 a^3\right ) \int \cos ^4(x) \, dx\\ &=\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x)+\frac {1}{8} \left (5 a^3\right ) \int \cos ^2(x) \, dx\\ &=\frac {5}{16} a^3 \cos (x) \sin (x)+\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x)+\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx\\ &=\frac {5 a^3 x}{16}+\frac {5}{16} a^3 \cos (x) \sin (x)+\frac {5}{24} a^3 \cos ^3(x) \sin (x)+\frac {1}{6} a^3 \cos ^5(x) \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 0.74 \[ a^3 \left (\frac {5 x}{16}+\frac {15}{64} \sin (2 x)+\frac {3}{64} \sin (4 x)+\frac {1}{192} \sin (6 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 37, normalized size = 0.80 \[ \frac {5}{16} \, a^{3} x + \frac {1}{48} \, {\left (8 \, a^{3} \cos \relax (x)^{5} + 10 \, a^{3} \cos \relax (x)^{3} + 15 \, a^{3} \cos \relax (x)\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 34, normalized size = 0.74 \[ \frac {5}{16} \, a^{3} x + \frac {1}{192} \, a^{3} \sin \left (6 \, x\right ) + \frac {3}{64} \, a^{3} \sin \left (4 \, x\right ) + \frac {15}{64} \, a^{3} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 72, normalized size = 1.57 \[ -a^{3} \left (-\frac {\left (\sin ^{5}\relax (x )+\frac {5 \left (\sin ^{3}\relax (x )\right )}{4}+\frac {15 \sin \relax (x )}{8}\right ) \cos \relax (x )}{6}+\frac {5 x}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\relax (x )+\frac {3 \sin \relax (x )}{2}\right ) \cos \relax (x )}{4}+\frac {3 x}{8}\right )-3 a^{3} \left (-\frac {\sin \relax (x ) \cos \relax (x )}{2}+\frac {x}{2}\right )+a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 69, normalized size = 1.50 \[ -\frac {1}{192} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{3} + \frac {3}{32} \, a^{3} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - \frac {3}{4} \, a^{3} {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.68, size = 42, normalized size = 0.91 \[ \frac {11\,a^3\,{\cos \relax (x)}^5\,\sin \relax (x)}{16}+\frac {5\,a^3\,{\cos \relax (x)}^3\,{\sin \relax (x)}^3}{6}+\frac {5\,a^3\,\cos \relax (x)\,{\sin \relax (x)}^5}{16}+\frac {5\,x\,a^3}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.66, size = 233, normalized size = 5.07 \[ - \frac {5 a^{3} x \sin ^{6}{\relax (x )}}{16} - \frac {15 a^{3} x \sin ^{4}{\relax (x )} \cos ^{2}{\relax (x )}}{16} + \frac {9 a^{3} x \sin ^{4}{\relax (x )}}{8} - \frac {15 a^{3} x \sin ^{2}{\relax (x )} \cos ^{4}{\relax (x )}}{16} + \frac {9 a^{3} x \sin ^{2}{\relax (x )} \cos ^{2}{\relax (x )}}{4} - \frac {3 a^{3} x \sin ^{2}{\relax (x )}}{2} - \frac {5 a^{3} x \cos ^{6}{\relax (x )}}{16} + \frac {9 a^{3} x \cos ^{4}{\relax (x )}}{8} - \frac {3 a^{3} x \cos ^{2}{\relax (x )}}{2} + a^{3} x + \frac {11 a^{3} \sin ^{5}{\relax (x )} \cos {\relax (x )}}{16} + \frac {5 a^{3} \sin ^{3}{\relax (x )} \cos ^{3}{\relax (x )}}{6} - \frac {15 a^{3} \sin ^{3}{\relax (x )} \cos {\relax (x )}}{8} + \frac {5 a^{3} \sin {\relax (x )} \cos ^{5}{\relax (x )}}{16} - \frac {9 a^{3} \sin {\relax (x )} \cos ^{3}{\relax (x )}}{8} + \frac {3 a^{3} \sin {\relax (x )} \cos {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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