Optimal. Leaf size=26 \[ \frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
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Rubi [A] time = 0.0313192, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2749} \[ \frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 2749
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 \left (-\frac{4 B}{5}+B \cos (c+d x)\right ) \, dx &=\frac{B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.184874, size = 31, normalized size = 1.19 \[ \frac{a^4 B \sin ^9(c+d x) \csc ^8\left (\frac{1}{2} (c+d x)\right )}{80 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 150, normalized size = 5.8 \begin{align*}{\frac{1}{5\,d} \left ({a}^{4}B \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sin \left ( dx+c \right ) +16\,{a}^{4}B \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{14\,{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}-4\,{a}^{4}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) -11\,{a}^{4}B\sin \left ( dx+c \right ) -4\,{a}^{4}B \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0555, size = 194, normalized size = 7.46 \begin{align*} \frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 28 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 6 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 24 \,{\left (d x + c\right )} B a^{4} - 66 \, B a^{4} \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.48511, size = 167, normalized size = 6.42 \begin{align*} \frac{{\left (B a^{4} \cos \left (d x + c\right )^{4} + 4 \, B a^{4} \cos \left (d x + c\right )^{3} + 6 \, B a^{4} \cos \left (d x + c\right )^{2} + 4 \, B a^{4} \cos \left (d x + c\right ) + B a^{4}\right )} \sin \left (d x + c\right )}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.54753, size = 333, normalized size = 12.81 \begin{align*} \begin{cases} \frac{6 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{5} + \frac{12 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5} - \frac{2 B a^{4} x \sin ^{2}{\left (c + d x \right )}}{5} + \frac{6 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{5} - \frac{2 B a^{4} x \cos ^{2}{\left (c + d x \right )}}{5} - \frac{4 B a^{4} x}{5} + \frac{8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{6 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{5 d} + \frac{28 B a^{4} \sin ^{3}{\left (c + d x \right )}}{15 d} + \frac{B a^{4} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{2 B a^{4} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} + \frac{14 B a^{4} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} - \frac{2 B a^{4} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{5 d} - \frac{11 B a^{4} \sin{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (B \cos{\left (c \right )} - \frac{4 B}{5}\right ) \left (a \cos{\left (c \right )} + a\right )^{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.82523, size = 119, normalized size = 4.58 \begin{align*} \frac{B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{B a^{4} \sin \left (4 \, d x + 4 \, c\right )}{10 \, d} + \frac{27 \, B a^{4} \sin \left (3 \, d x + 3 \, c\right )}{80 \, d} + \frac{3 \, B a^{4} \sin \left (2 \, d x + 2 \, c\right )}{5 \, d} + \frac{21 \, B a^{4} \sin \left (d x + c\right )}{40 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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