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 = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 8
Rule 2635
Rule 3175
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.45, size = 46, normalized size = 0.78 \[ \frac {35}{128} \, a^{4} x + \frac {1}{384} \, {\left (48 \, a^{4} \cos \relax (x)^{7} + 56 \, a^{4} \cos \relax (x)^{5} + 70 \, a^{4} \cos \relax (x)^{3} + 105 \, a^{4} \cos \relax (x)\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 43, normalized size = 0.73 \[ \frac {35}{128} \, a^{4} x + \frac {1}{1024} \, a^{4} \sin \left (8 \, x\right ) + \frac {1}{96} \, a^{4} \sin \left (6 \, x\right ) + \frac {7}{128} \, a^{4} \sin \left (4 \, x\right ) + \frac {7}{32} \, a^{4} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 105, normalized size = 1.78 \[ a^{4} \left (-\frac {\left (\sin ^{7}\relax (x )+\frac {7 \left (\sin ^{5}\relax (x )\right )}{6}+\frac {35 \left (\sin ^{3}\relax (x )\right )}{24}+\frac {35 \sin \relax (x )}{16}\right ) \cos \relax (x )}{8}+\frac {35 x}{128}\right )-4 a^{4} \left (-\frac {\left (\sin ^{5}\relax (x )+\frac {5 \left (\sin ^{3}\relax (x )\right )}{4}+\frac {15 \sin \relax (x )}{8}\right ) \cos \relax (x )}{6}+\frac {5 x}{16}\right )+6 a^{4} \left (-\frac {\left (\sin ^{3}\relax (x )+\frac {3 \sin \relax (x )}{2}\right ) \cos \relax (x )}{4}+\frac {3 x}{8}\right )-4 a^{4} \left (-\frac {\sin \relax (x ) \cos \relax (x )}{2}+\frac {x}{2}\right )+a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 104, normalized size = 1.76 \[ \frac {1}{3072} \, {\left (128 \, \sin \left (2 \, x\right )^{3} + 840 \, x + 3 \, \sin \left (8 \, x\right ) + 168 \, \sin \left (4 \, x\right ) - 768 \, \sin \left (2 \, x\right )\right )} a^{4} - \frac {1}{48} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a^{4} + \frac {3}{16} \, a^{4} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - a^{4} {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.69, size = 51, normalized size = 0.86 \[ \frac {\frac {35\,a^4\,{\mathrm {tan}\relax (x)}^7}{128}+\frac {385\,a^4\,{\mathrm {tan}\relax (x)}^5}{384}+\frac {511\,a^4\,{\mathrm {tan}\relax (x)}^3}{384}+\frac {93\,a^4\,\mathrm {tan}\relax (x)}{128}}{{\left ({\mathrm {tan}\relax (x)}^2+1\right )}^4}+\frac {35\,a^4\,x}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.32, size = 376, normalized size = 6.37 \[ \frac {35 a^{4} x \sin ^{8}{\relax (x )}}{128} + \frac {35 a^{4} x \sin ^{6}{\relax (x )} \cos ^{2}{\relax (x )}}{32} - \frac {5 a^{4} x \sin ^{6}{\relax (x )}}{4} + \frac {105 a^{4} x \sin ^{4}{\relax (x )} \cos ^{4}{\relax (x )}}{64} - \frac {15 a^{4} x \sin ^{4}{\relax (x )} \cos ^{2}{\relax (x )}}{4} + \frac {9 a^{4} x \sin ^{4}{\relax (x )}}{4} + \frac {35 a^{4} x \sin ^{2}{\relax (x )} \cos ^{6}{\relax (x )}}{32} - \frac {15 a^{4} x \sin ^{2}{\relax (x )} \cos ^{4}{\relax (x )}}{4} + \frac {9 a^{4} x \sin ^{2}{\relax (x )} \cos ^{2}{\relax (x )}}{2} - 2 a^{4} x \sin ^{2}{\relax (x )} + \frac {35 a^{4} x \cos ^{8}{\relax (x )}}{128} - \frac {5 a^{4} x \cos ^{6}{\relax (x )}}{4} + \frac {9 a^{4} x \cos ^{4}{\relax (x )}}{4} - 2 a^{4} x \cos ^{2}{\relax (x )} + a^{4} x - \frac {93 a^{4} \sin ^{7}{\relax (x )} \cos {\relax (x )}}{128} - \frac {511 a^{4} \sin ^{5}{\relax (x )} \cos ^{3}{\relax (x )}}{384} + \frac {11 a^{4} \sin ^{5}{\relax (x )} \cos {\relax (x )}}{4} - \frac {385 a^{4} \sin ^{3}{\relax (x )} \cos ^{5}{\relax (x )}}{384} + \frac {10 a^{4} \sin ^{3}{\relax (x )} \cos ^{3}{\relax (x )}}{3} - \frac {15 a^{4} \sin ^{3}{\relax (x )} \cos {\relax (x )}}{4} - \frac {35 a^{4} \sin {\relax (x )} \cos ^{7}{\relax (x )}}{128} + \frac {5 a^{4} \sin {\relax (x )} \cos ^{5}{\relax (x )}}{4} - \frac {9 a^{4} \sin {\relax (x )} \cos ^{3}{\relax (x )}}{4} + 2 a^{4} \sin {\relax (x )} \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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