Optimal. Leaf size=88 \[ \frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac {7 \sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac {35 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac {35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac {35 x}{128} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2635, 8} \[ \frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac {7 \sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac {35 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac {35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac {35 x}{128} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \cos ^8(a+b x) \, dx &=\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {7}{8} \int \cos ^6(a+b x) \, dx\\ &=\frac {7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {35}{48} \int \cos ^4(a+b x) \, dx\\ &=\frac {35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {35}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac {35 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac {35 \int 1 \, dx}{128}\\ &=\frac {35 x}{128}+\frac {35 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac {7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac {\cos ^7(a+b x) \sin (a+b x)}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.62 \[ \frac {672 \sin (2 (a+b x))+168 \sin (4 (a+b x))+32 \sin (6 (a+b x))+3 \sin (8 (a+b x))+840 a+840 b x}{3072 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 56, normalized size = 0.64 \[ \frac {105 \, b x + {\left (48 \, \cos \left (b x + a\right )^{7} + 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} + 105 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 60, normalized size = 0.68 \[ \frac {35}{128} \, x + \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} + \frac {\sin \left (6 \, b x + 6 \, a\right )}{96 \, b} + \frac {7 \, \sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac {7 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 58, normalized size = 0.66 \[ \frac {\frac {\left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )}{8}+\frac {35 b x}{128}+\frac {35 a}{128}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 59, normalized size = 0.67 \[ -\frac {128 \, \sin \left (2 \, b x + 2 \, a\right )^{3} - 840 \, b x - 840 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 168 \, \sin \left (4 \, b x + 4 \, a\right ) - 768 \, \sin \left (2 \, b x + 2 \, a\right )}{3072 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 53, normalized size = 0.60 \[ \frac {35\,x}{128}+\frac {\frac {7\,\sin \left (2\,a+2\,b\,x\right )}{32}+\frac {7\,\sin \left (4\,a+4\,b\,x\right )}{128}+\frac {\sin \left (6\,a+6\,b\,x\right )}{96}+\frac {\sin \left (8\,a+8\,b\,x\right )}{1024}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.67, size = 184, normalized size = 2.09 \[ \begin {cases} \frac {35 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {35 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {105 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {35 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {35 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac {35 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} + \frac {385 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac {511 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} + \frac {93 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \cos ^{8}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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