Optimal. Leaf size=130 \[ -\frac{3 a^3 \cos ^5(c+d x)}{10 d}-\frac{3 \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{14 d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{9 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9 a^3 x}{16}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
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Rubi [A] time = 0.140113, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2678, 2669, 2635, 8} \[ -\frac{3 a^3 \cos ^5(c+d x)}{10 d}-\frac{3 \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{14 d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{9 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9 a^3 x}{16}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac{1}{7} (9 a) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac{3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac{1}{2} \left (3 a^2\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{3 a^3 \cos ^5(c+d x)}{10 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac{3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac{1}{2} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{3 a^3 \cos ^5(c+d x)}{10 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac{3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac{1}{8} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{3 a^3 \cos ^5(c+d x)}{10 d}+\frac{9 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac{3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac{1}{16} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac{9 a^3 x}{16}-\frac{3 a^3 \cos ^5(c+d x)}{10 d}+\frac{9 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac{a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac{3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}\\ \end{align*}
Mathematica [A] time = 0.793144, size = 161, normalized size = 1.24 \[ -\frac{a^3 \left (\sqrt{\sin (c+d x)+1} \left (80 \sin ^7(c+d x)+200 \sin ^6(c+d x)-72 \sin ^5(c+d x)-558 \sin ^4(c+d x)-306 \sin ^3(c+d x)+411 \sin ^2(c+d x)+613 \sin (c+d x)-368\right )-630 \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right ) \sqrt{1-\sin (c+d x)}\right ) \cos ^5(c+d x)}{560 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 143, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +3\,{a}^{3} \left ( -1/6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) -{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5}}+{a}^{3} \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 = 0.966801, size = 155, normalized size = 1.19 \begin{align*} -\frac{1344 \, a^{3} \cos \left (d x + c\right )^{5} - 64 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 35 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 70 \,{\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^{3}}{2240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76231, size = 212, normalized size = 1.63 \begin{align*} \frac{80 \, a^{3} \cos \left (d x + c\right )^{7} - 448 \, a^{3} \cos \left (d x + c\right )^{5} + 315 \, a^{3} d x - 35 \,{\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 6 \, a^{3} \cos \left (d x + c\right )^{3} - 9 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{560 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.62774, size = 335, normalized size = 2.58 \begin{align*} \begin{cases} \frac{3 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{3 a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{3 a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac{3 a^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} - \frac{a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{3 a^{3} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac{5 a^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac{2 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac{3 a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \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 = 1.16627, size = 166, normalized size = 1.28 \begin{align*} \frac{9}{16} \, a^{3} x + \frac{a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{11 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{13 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{27 \, a^{3} \cos \left (d x + c\right )}{64 \, d} - \frac{a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac{a^{3} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{19 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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