Optimal. Leaf size=54 \[ -\frac{\sin ^7(a+b x)}{7 b}+\frac{3 \sin ^5(a+b x)}{5 b}-\frac{\sin ^3(a+b x)}{b}+\frac{\sin (a+b x)}{b} \]
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Rubi [A] time = 0.0163204, 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} \[ -\frac{\sin ^7(a+b x)}{7 b}+\frac{3 \sin ^5(a+b x)}{5 b}-\frac{\sin ^3(a+b x)}{b}+\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \cos ^7(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{b}+\frac{3 \sin ^5(a+b x)}{5 b}-\frac{\sin ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0132348, size = 54, normalized size = 1. \[ -\frac{\sin ^7(a+b x)}{7 b}+\frac{3 \sin ^5(a+b x)}{5 b}-\frac{\sin ^3(a+b x)}{b}+\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 42, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( bx+a \right ) }{7\,b} \left ({\frac{16}{5}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \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 = 1.63903, size = 59, normalized size = 1.09 \begin{align*} -\frac{5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \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 = 1.68887, size = 112, normalized size = 2.07 \begin{align*} \frac{{\left (5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \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 = 6.8842, size = 78, normalized size = 1.44 \begin{align*} \begin{cases} \frac{16 \sin ^{7}{\left (a + b x \right )}}{35 b} + \frac{8 \sin ^{5}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{5 b} + \frac{2 \sin ^{3}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{b} + \frac{\sin{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cos ^{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 = 1.51886, size = 59, normalized size = 1.09 \begin{align*} -\frac{5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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