Optimal. Leaf size=60 \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.0422384, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2668, 641, 194} \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 641
Rule 194
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a+x) \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac{b \cos ^6(c+d x)}{6 d}+\frac{a \operatorname{Subst}\left (\int \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac{b \cos ^6(c+d x)}{6 d}+\frac{a \operatorname{Subst}\left (\int \left (b^4-2 b^2 x^2+x^4\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac{b \cos ^6(c+d x)}{6 d}+\frac{a \sin (c+d x)}{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 = 0.0232151, size = 60, normalized size = 1. \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 46, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}+{\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 ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948505, size = 95, normalized size = 1.58 \begin{align*} \frac{5 \, b \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 15 \, b \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} + 15 \, b \sin \left (d x + c\right )^{2} + 30 \, a \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 [A] time = 2.3126, size = 128, normalized size = 2.13 \begin{align*} -\frac{5 \, b \cos \left (d x + c\right )^{6} - 2 \,{\left (3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.76841, size = 83, normalized size = 1.38 \begin{align*} \begin{cases} \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{a \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{b \cos ^{6}{\left (c + d x \right )}}{6 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right ) \cos ^{5}{\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 = 1.0811, size = 119, normalized size = 1.98 \begin{align*} -\frac{b \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{b \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac{5 \, b \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{5 \, a \sin \left (3 \, d x + 3 \, c\right )}{48 \, 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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