Optimal. Leaf size=87 \[ -\frac{a \cos ^5(c+d x)}{5 d}-\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b x}{16} \]
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Rubi [A] time = 0.110956, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2838, 2565, 30, 2568, 2635, 8} \[ -\frac{a \cos ^5(c+d x)}{5 d}-\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b x}{16} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2565
Rule 30
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^4(c+d x) \sin (c+d x) \, dx+b \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac{b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{6} b \int \cos ^4(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \cos ^5(c+d x)}{5 d}+\frac{b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac{b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{8} b \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^5(c+d x)}{5 d}+\frac{b \cos (c+d x) \sin (c+d x)}{16 d}+\frac{b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac{b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{16} b \int 1 \, dx\\ &=\frac{b x}{16}-\frac{a \cos ^5(c+d x)}{5 d}+\frac{b \cos (c+d x) \sin (c+d x)}{16 d}+\frac{b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac{b \cos ^5(c+d x) \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.161803, size = 77, normalized size = 0.89 \[ -\frac{120 a \cos (c+d x)+60 a \cos (3 (c+d x))+12 a \cos (5 (c+d x))-15 b \sin (2 (c+d x))+15 b \sin (4 (c+d x))+5 b \sin (6 (c+d x))-60 b d x}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 68, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5}}+b \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{\sin \left ( dx+c \right ) }{24} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{dx}{16}}+{\frac{c}{16}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10714, size = 70, normalized size = 0.8 \begin{align*} -\frac{192 \, a \cos \left (d x + c\right )^{5} - 5 \,{\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 )} b}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37966, size = 163, normalized size = 1.87 \begin{align*} -\frac{48 \, a \cos \left (d x + c\right )^{5} - 15 \, b d x + 5 \,{\left (8 \, b \cos \left (d x + c\right )^{5} - 2 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.68937, size = 167, normalized size = 1.92 \begin{align*} \begin{cases} - \frac{a \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac{b x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{3 b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{b x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{b \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac{b \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right ) \sin{\left (c \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 = 1.31486, size = 124, normalized size = 1.43 \begin{align*} \frac{1}{16} \, b x - \frac{a \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac{a \cos \left (d x + c\right )}{8 \, d} - \frac{b \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{b \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{b \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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