Optimal. Leaf size=54 \[ \frac{\cos ^7(a+b x)}{7 b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^3(a+b x)}{b}-\frac{\cos (a+b x)}{b} \]
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Rubi [A] time = 0.0157233, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ \frac{\cos ^7(a+b x)}{7 b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^3(a+b x)}{b}-\frac{\cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \sin ^7(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos (a+b x)}{b}+\frac{\cos ^3(a+b x)}{b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0092732, size = 59, normalized size = 1.09 \[ -\frac{35 \cos (a+b x)}{64 b}+\frac{7 \cos (3 (a+b x))}{64 b}-\frac{7 \cos (5 (a+b x))}{320 b}+\frac{\cos (7 (a+b x))}{448 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 42, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{7\,b} \left ({\frac{16}{5}}+ \left ( \sin \left ( bx+a \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979021, size = 59, normalized size = 1.09 \begin{align*} \frac{5 \, \cos \left (b x + a\right )^{7} - 21 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27506, size = 115, normalized size = 2.13 \begin{align*} \frac{5 \, \cos \left (b x + a\right )^{7} - 21 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.82139, size = 80, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{\sin ^{6}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{2 \sin ^{4}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac{8 \sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac{16 \cos ^{7}{\left (a + b x \right )}}{35 b} & \text{for}\: b \neq 0 \\x \sin ^{7}{\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 = 1.13322, size = 68, normalized size = 1.26 \begin{align*} \frac{\cos \left (b x + a\right )^{7}}{7 \, b} - \frac{3 \, \cos \left (b x + a\right )^{5}}{5 \, b} + \frac{\cos \left (b x + a\right )^{3}}{b} - \frac{\cos \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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