Optimal. Leaf size=59 \[ \frac{35 a^4 x}{128}+\frac{1}{8} a^4 \sin (x) \cos ^7(x)+\frac{7}{48} a^4 \sin (x) \cos ^5(x)+\frac{35}{192} a^4 \sin (x) \cos ^3(x)+\frac{35}{128} a^4 \sin (x) \cos (x) \]
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Rubi [A] time = 0.0415468, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3175, 2635, 8} \[ \frac{35 a^4 x}{128}+\frac{1}{8} a^4 \sin (x) \cos ^7(x)+\frac{7}{48} a^4 \sin (x) \cos ^5(x)+\frac{35}{192} a^4 \sin (x) \cos ^3(x)+\frac{35}{128} a^4 \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a-a \sin ^2(x)\right )^4 \, dx &=a^4 \int \cos ^8(x) \, dx\\ &=\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{8} \left (7 a^4\right ) \int \cos ^6(x) \, dx\\ &=\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{48} \left (35 a^4\right ) \int \cos ^4(x) \, dx\\ &=\frac{35}{192} a^4 \cos ^3(x) \sin (x)+\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{64} \left (35 a^4\right ) \int \cos ^2(x) \, dx\\ &=\frac{35}{128} a^4 \cos (x) \sin (x)+\frac{35}{192} a^4 \cos ^3(x) \sin (x)+\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)+\frac{1}{128} \left (35 a^4\right ) \int 1 \, dx\\ &=\frac{35 a^4 x}{128}+\frac{35}{128} a^4 \cos (x) \sin (x)+\frac{35}{192} a^4 \cos ^3(x) \sin (x)+\frac{7}{48} a^4 \cos ^5(x) \sin (x)+\frac{1}{8} a^4 \cos ^7(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0030452, size = 42, normalized size = 0.71 \[ a^4 \left (\frac{35 x}{128}+\frac{7}{32} \sin (2 x)+\frac{7}{128} \sin (4 x)+\frac{1}{96} \sin (6 x)+\frac{\sin (8 x)}{1024}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 105, normalized size = 1.8 \begin{align*}{a}^{4} \left ( -{\frac{\cos \left ( x \right ) }{8} \left ( \left ( \sin \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{128}} \right ) -4\,{a}^{4} \left ( -1/6\, \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+5/4\, \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) \cos \left ( x \right ) +{\frac{5\,x}{16}} \right ) +6\,{a}^{4} \left ( -1/4\, \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+3/2\,\sin \left ( x \right ) \right ) \cos \left ( x \right ) +3/8\,x \right ) -4\,{a}^{4} \left ( -1/2\,\sin \left ( x \right ) \cos \left ( x \right ) +x/2 \right ) +{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.958191, size = 140, normalized size = 2.37 \begin{align*} \frac{1}{3072} \,{\left (128 \, \sin \left (2 \, x\right )^{3} + 840 \, x + 3 \, \sin \left (8 \, x\right ) + 168 \, \sin \left (4 \, x\right ) - 768 \, \sin \left (2 \, x\right )\right )} a^{4} - \frac{1}{48} \,{\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{4} + \frac{3}{16} \, a^{4}{\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - a^{4}{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74637, size = 135, normalized size = 2.29 \begin{align*} \frac{35}{128} \, a^{4} x + \frac{1}{384} \,{\left (48 \, a^{4} \cos \left (x\right )^{7} + 56 \, a^{4} \cos \left (x\right )^{5} + 70 \, a^{4} \cos \left (x\right )^{3} + 105 \, a^{4} \cos \left (x\right )\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.8102, size = 376, normalized size = 6.37 \begin{align*} \frac{35 a^{4} x \sin ^{8}{\left (x \right )}}{128} + \frac{35 a^{4} x \sin ^{6}{\left (x \right )} \cos ^{2}{\left (x \right )}}{32} - \frac{5 a^{4} x \sin ^{6}{\left (x \right )}}{4} + \frac{105 a^{4} x \sin ^{4}{\left (x \right )} \cos ^{4}{\left (x \right )}}{64} - \frac{15 a^{4} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac{9 a^{4} x \sin ^{4}{\left (x \right )}}{4} + \frac{35 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{6}{\left (x \right )}}{32} - \frac{15 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{4} + \frac{9 a^{4} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} - 2 a^{4} x \sin ^{2}{\left (x \right )} + \frac{35 a^{4} x \cos ^{8}{\left (x \right )}}{128} - \frac{5 a^{4} x \cos ^{6}{\left (x \right )}}{4} + \frac{9 a^{4} x \cos ^{4}{\left (x \right )}}{4} - 2 a^{4} x \cos ^{2}{\left (x \right )} + a^{4} x - \frac{93 a^{4} \sin ^{7}{\left (x \right )} \cos{\left (x \right )}}{128} - \frac{511 a^{4} \sin ^{5}{\left (x \right )} \cos ^{3}{\left (x \right )}}{384} + \frac{11 a^{4} \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{4} - \frac{385 a^{4} \sin ^{3}{\left (x \right )} \cos ^{5}{\left (x \right )}}{384} + \frac{10 a^{4} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{3} - \frac{15 a^{4} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{4} - \frac{35 a^{4} \sin{\left (x \right )} \cos ^{7}{\left (x \right )}}{128} + \frac{5 a^{4} \sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{4} - \frac{9 a^{4} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + 2 a^{4} \sin{\left (x \right )} \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13789, size = 58, normalized size = 0.98 \begin{align*} \frac{35}{128} \, a^{4} x + \frac{1}{1024} \, a^{4} \sin \left (8 \, x\right ) + \frac{1}{96} \, a^{4} \sin \left (6 \, x\right ) + \frac{7}{128} \, a^{4} \sin \left (4 \, x\right ) + \frac{7}{32} \, a^{4} \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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