Optimal. Leaf size=92 \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8} \]
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Rubi [A] time = 0.0575704, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2748, 2635, 8, 2633} \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx &=a \int \cos ^4(c+d x) \, dx+a \int \cos ^5(c+d x) \, dx\\ &=\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} (3 a) \int \cos ^2(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{a \sin (c+d x)}{d}+\frac{3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{1}{8} (3 a) \int 1 \, dx\\ &=\frac{3 a x}{8}+\frac{a \sin (c+d x)}{d}+\frac{3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.107791, size = 65, normalized size = 0.71 \[ \frac{a \left (96 \sin ^5(c+d x)-320 \sin ^3(c+d x)+480 \sin (c+d x)+15 (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))\right )}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 70, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{a\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+a \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10983, size = 93, normalized size = 1.01 \begin{align*} \frac{32 \,{\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 )} a + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95596, size = 173, normalized size = 1.88 \begin{align*} \frac{45 \, a d x +{\left (24 \, a \cos \left (d x + c\right )^{4} + 30 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} + 45 \, a \cos \left (d x + c\right ) + 64 \, a\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.38951, size = 168, normalized size = 1.83 \begin{align*} \begin{cases} \frac{3 a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{8 a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 a \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{a \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{5 a \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right ) \cos ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38839, size = 104, normalized size = 1.13 \begin{align*} \frac{3}{8} \, a x + \frac{a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{5 \, a \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{5 \, a \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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