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 = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.21, 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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fricas [B] time = 0.82, size = 70, normalized size = 2.69 \[ \frac {{\left (B a^{4} \cos \left (d x + c\right )^{4} + 4 \, B a^{4} \cos \left (d x + c\right )^{3} + 6 \, B a^{4} \cos \left (d x + c\right )^{2} + 4 \, B a^{4} \cos \left (d x + c\right ) + B a^{4}\right )} \sin \left (d x + c\right )}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 88, normalized size = 3.38 \[ \frac {B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {B a^{4} \sin \left (4 \, d x + 4 \, c\right )}{10 \, d} + \frac {27 \, B a^{4} \sin \left (3 \, d x + 3 \, c\right )}{80 \, d} + \frac {3 \, B a^{4} \sin \left (2 \, d x + 2 \, c\right )}{5 \, d} + \frac {21 \, B a^{4} \sin \left (d x + c\right )}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 150, normalized size = 5.77 \[ \frac {a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )+16 a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {14 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}-4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-11 a^{4} B \sin \left (d x +c \right )-4 a^{4} B \left (d x +c \right )}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 144, normalized size = 5.54 \[ \frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 28 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 24 \, {\left (d x + c\right )} B a^{4} - 66 \, B a^{4} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 29, normalized size = 1.12 \[ \frac {32\,B\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.46, size = 333, normalized size = 12.81 \[ \begin {cases} \frac {6 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{5} + \frac {12 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5} - \frac {2 B a^{4} x \sin ^{2}{\left (c + d x \right )}}{5} + \frac {6 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{5} - \frac {2 B a^{4} x \cos ^{2}{\left (c + d x \right )}}{5} - \frac {4 B a^{4} x}{5} + \frac {8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {6 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{5 d} + \frac {28 B a^{4} \sin ^{3}{\left (c + d x \right )}}{15 d} + \frac {B a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} + \frac {14 B a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} - \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{5 d} - \frac {11 B a^{4} \sin {\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (B \cos {\relax (c )} - \frac {4 B}{5}\right ) \left (a \cos {\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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