Optimal. Leaf size=88 \[ -\frac{\sin ^7(a+b x) \cos (a+b x)}{8 b}-\frac{7 \sin ^5(a+b x) \cos (a+b x)}{48 b}-\frac{35 \sin ^3(a+b x) \cos (a+b x)}{192 b}-\frac{35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{35 x}{128} \]
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Rubi [A] time = 0.0484121, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ -\frac{\sin ^7(a+b x) \cos (a+b x)}{8 b}-\frac{7 \sin ^5(a+b x) \cos (a+b x)}{48 b}-\frac{35 \sin ^3(a+b x) \cos (a+b x)}{192 b}-\frac{35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{35 x}{128} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sin ^8(a+b x) \, dx &=-\frac{\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac{7}{8} \int \sin ^6(a+b x) \, dx\\ &=-\frac{7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac{\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac{35}{48} \int \sin ^4(a+b x) \, dx\\ &=-\frac{35 \cos (a+b x) \sin ^3(a+b x)}{192 b}-\frac{7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac{\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac{35}{64} \int \sin ^2(a+b x) \, dx\\ &=-\frac{35 \cos (a+b x) \sin (a+b x)}{128 b}-\frac{35 \cos (a+b x) \sin ^3(a+b x)}{192 b}-\frac{7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac{\cos (a+b x) \sin ^7(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 (a+b x) \sin ^3(a+b x)}{192 b}-\frac{7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac{\cos (a+b x) \sin ^7(a+b x)}{8 b}\\ \end{align*}
Mathematica [A] time = 0.0557637, 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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Maple [A] time = 0.034, size = 58, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{\cos \left ( bx+a \right ) }{8} \left ( \left ( \sin \left ( bx+a \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( bx+a \right ) }{16}} \right ) }+{\frac{35\,bx}{128}}+{\frac{35\,a}{128}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02275, size = 80, normalized size = 0.91 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27506, size = 155, normalized size = 1.76 \begin{align*} \frac{105 \, b x +{\left (48 \, \cos \left (b x + a\right )^{7} - 200 \, \cos \left (b x + a\right )^{5} + 326 \, \cos \left (b x + a\right )^{3} - 279 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.3102, size = 184, normalized size = 2.09 \begin{align*} \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{93 \sin ^{7}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{128 b} - \frac{511 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} - \frac{385 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac{35 \sin{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sin ^{8}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13179, size = 81, normalized size = 0.92 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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