Optimal. Leaf size=109 \[ \frac{d \cos (a+b x)}{8 b^2}-\frac{d \cos (3 a+3 b x)}{144 b^2}-\frac{d \cos (5 a+5 b x)}{400 b^2}+\frac{(c+d x) \sin (a+b x)}{8 b}-\frac{(c+d x) \sin (3 a+3 b x)}{48 b}-\frac{(c+d x) \sin (5 a+5 b x)}{80 b} \]
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Rubi [A] time = 0.0939269, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4406, 3296, 2638} \[ \frac{d \cos (a+b x)}{8 b^2}-\frac{d \cos (3 a+3 b x)}{144 b^2}-\frac{d \cos (5 a+5 b x)}{400 b^2}+\frac{(c+d x) \sin (a+b x)}{8 b}-\frac{(c+d x) \sin (3 a+3 b x)}{48 b}-\frac{(c+d x) \sin (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 4406
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x) \cos (a+b x)-\frac{1}{16} (c+d x) \cos (3 a+3 b x)-\frac{1}{16} (c+d x) \cos (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac{1}{16} \int (c+d x) \cos (3 a+3 b x) \, dx\right )-\frac{1}{16} \int (c+d x) \cos (5 a+5 b x) \, dx+\frac{1}{8} \int (c+d x) \cos (a+b x) \, dx\\ &=\frac{(c+d x) \sin (a+b x)}{8 b}-\frac{(c+d x) \sin (3 a+3 b x)}{48 b}-\frac{(c+d x) \sin (5 a+5 b x)}{80 b}+\frac{d \int \sin (5 a+5 b x) \, dx}{80 b}+\frac{d \int \sin (3 a+3 b x) \, dx}{48 b}-\frac{d \int \sin (a+b x) \, dx}{8 b}\\ &=\frac{d \cos (a+b x)}{8 b^2}-\frac{d \cos (3 a+3 b x)}{144 b^2}-\frac{d \cos (5 a+5 b x)}{400 b^2}+\frac{(c+d x) \sin (a+b x)}{8 b}-\frac{(c+d x) \sin (3 a+3 b x)}{48 b}-\frac{(c+d x) \sin (5 a+5 b x)}{80 b}\\ \end{align*}
Mathematica [A] time = 0.365338, size = 110, normalized size = 1.01 \[ -\frac{-450 b c \sin (a+b x)+75 b c \sin (3 (a+b x))+45 b c \sin (5 (a+b x))-450 b d x \sin (a+b x)+75 b d x \sin (3 (a+b x))+45 b d x \sin (5 (a+b x))-450 d \cos (a+b x)+25 d \cos (3 (a+b x))+9 d \cos (5 (a+b x))}{3600 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 175, normalized size = 1.6 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ({\frac{ \left ( bx+a \right ) \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{45}}+{\frac{2\,\cos \left ( bx+a \right ) }{15}}-{\frac{ \left ( bx+a \right ) \sin \left ( bx+a \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) }-{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{25}} \right ) }-{\frac{ad}{b} \left ( -{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{ \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{15}} \right ) }+c \left ( -{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{ \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{15}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05287, size = 188, normalized size = 1.72 \begin{align*} -\frac{240 \,{\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )} c - \frac{240 \,{\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )} a d}{b} + \frac{{\left (45 \,{\left (b x + a\right )} \sin \left (5 \, b x + 5 \, a\right ) + 75 \,{\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 450 \,{\left (b x + a\right )} \sin \left (b x + a\right ) + 9 \, \cos \left (5 \, b x + 5 \, a\right ) + 25 \, \cos \left (3 \, b x + 3 \, a\right ) - 450 \, \cos \left (b x + a\right )\right )} d}{b}}{3600 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.488231, size = 235, normalized size = 2.16 \begin{align*} -\frac{9 \, d \cos \left (b x + a\right )^{5} - 5 \, d \cos \left (b x + a\right )^{3} - 30 \, d \cos \left (b x + a\right ) + 15 \,{\left (3 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{4} - 2 \, b d x -{\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} - 2 \, b c\right )} \sin \left (b x + a\right )}{225 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.46708, size = 163, normalized size = 1.5 \begin{align*} \begin{cases} \frac{2 c \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac{c \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac{2 d x \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac{d x \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac{2 d \sin ^{4}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{15 b^{2}} + \frac{13 d \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{45 b^{2}} + \frac{26 d \cos ^{5}{\left (a + b x \right )}}{225 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin ^{2}{\left (a \right )} \cos ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11593, size = 143, normalized size = 1.31 \begin{align*} -\frac{d \cos \left (5 \, b x + 5 \, a\right )}{400 \, b^{2}} - \frac{d \cos \left (3 \, b x + 3 \, a\right )}{144 \, b^{2}} + \frac{d \cos \left (b x + a\right )}{8 \, b^{2}} - \frac{{\left (b d x + b c\right )} \sin \left (5 \, b x + 5 \, a\right )}{80 \, b^{2}} - \frac{{\left (b d x + b c\right )} \sin \left (3 \, b x + 3 \, a\right )}{48 \, b^{2}} + \frac{{\left (b d x + b c\right )} \sin \left (b x + a\right )}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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