Optimal. Leaf size=134 \[ \frac{d (c+d x) \sin (a+b x) \cos ^3(a+b x)}{8 b^2}+\frac{3 d (c+d x) \sin (a+b x) \cos (a+b x)}{16 b^2}+\frac{d^2 \cos ^4(a+b x)}{32 b^3}+\frac{3 d^2 \cos ^2(a+b x)}{32 b^3}-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{3 c d x}{16 b}+\frac{3 d^2 x^2}{32 b} \]
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Rubi [A] time = 0.086674, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4405, 3310} \[ \frac{d (c+d x) \sin (a+b x) \cos ^3(a+b x)}{8 b^2}+\frac{3 d (c+d x) \sin (a+b x) \cos (a+b x)}{16 b^2}+\frac{d^2 \cos ^4(a+b x)}{32 b^3}+\frac{3 d^2 \cos ^2(a+b x)}{32 b^3}-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{3 c d x}{16 b}+\frac{3 d^2 x^2}{32 b} \]
Antiderivative was successfully verified.
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Rule 4405
Rule 3310
Rubi steps
\begin{align*} \int (c+d x)^2 \cos ^3(a+b x) \sin (a+b x) \, dx &=-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{d \int (c+d x) \cos ^4(a+b x) \, dx}{2 b}\\ &=\frac{d^2 \cos ^4(a+b x)}{32 b^3}-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{d (c+d x) \cos ^3(a+b x) \sin (a+b x)}{8 b^2}+\frac{(3 d) \int (c+d x) \cos ^2(a+b x) \, dx}{8 b}\\ &=\frac{3 d^2 \cos ^2(a+b x)}{32 b^3}+\frac{d^2 \cos ^4(a+b x)}{32 b^3}-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{3 d (c+d x) \cos (a+b x) \sin (a+b x)}{16 b^2}+\frac{d (c+d x) \cos ^3(a+b x) \sin (a+b x)}{8 b^2}+\frac{(3 d) \int (c+d x) \, dx}{16 b}\\ &=\frac{3 c d x}{16 b}+\frac{3 d^2 x^2}{32 b}+\frac{3 d^2 \cos ^2(a+b x)}{32 b^3}+\frac{d^2 \cos ^4(a+b x)}{32 b^3}-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{3 d (c+d x) \cos (a+b x) \sin (a+b x)}{16 b^2}+\frac{d (c+d x) \cos ^3(a+b x) \sin (a+b x)}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.486177, size = 89, normalized size = 0.66 \[ \frac{-16 \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-d^2\right )+\cos (4 (a+b x)) \left (d^2-8 b^2 (c+d x)^2\right )+4 b d (c+d x) (8 \sin (2 (a+b x))+\sin (4 (a+b x)))}{256 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 260, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2}}{{b}^{2}} \left ( -{\frac{ \left ( bx+a \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{4}}+{\frac{bx+a}{2} \left ({\frac{\sin \left ( bx+a \right ) }{4} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+{\frac{3\,\cos \left ( bx+a \right ) }{2}} \right ) }+{\frac{3\,bx}{8}}+{\frac{3\,a}{8}} \right ) }-{\frac{3\, \left ( bx+a \right ) ^{2}}{32}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{32}}+{\frac{3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{32}} \right ) }-2\,{\frac{a{d}^{2}}{{b}^{2}} \left ( -1/4\, \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}+1/16\, \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+3/2\,\cos \left ( bx+a \right ) \right ) \sin \left ( bx+a \right ) +{\frac{3\,bx}{32}}+{\frac{3\,a}{32}} \right ) }+2\,{\frac{cd}{b} \left ( -1/4\, \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}+1/16\, \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+3/2\,\cos \left ( bx+a \right ) \right ) \sin \left ( bx+a \right ) +{\frac{3\,bx}{32}}+{\frac{3\,a}{32}} \right ) }-{\frac{{a}^{2}{d}^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{4\,{b}^{2}}}+{\frac{cda \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{2\,b}}-{\frac{{c}^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15485, size = 355, normalized size = 2.65 \begin{align*} -\frac{64 \, c^{2} \cos \left (b x + a\right )^{4} - \frac{128 \, a c d \cos \left (b x + a\right )^{4}}{b} + \frac{64 \, a^{2} d^{2} \cos \left (b x + a\right )^{4}}{b^{2}} + \frac{4 \,{\left (4 \,{\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c d}{b} - \frac{4 \,{\left (4 \,{\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{2}}{b^{2}} + \frac{{\left ({\left (8 \,{\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \,{\left (2 \,{\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 4 \,{\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - 32 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{2}}{b^{2}}}{256 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.49275, size = 294, normalized size = 2.19 \begin{align*} \frac{3 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c d x -{\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{4} + 3 \, d^{2} \cos \left (b x + a\right )^{2} + 2 \,{\left (2 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} + 3 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{32 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.5359, size = 350, normalized size = 2.61 \begin{align*} \begin{cases} \frac{c^{2} \sin ^{4}{\left (a + b x \right )}}{4 b} + \frac{c^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{2 b} + \frac{3 c d x \sin ^{4}{\left (a + b x \right )}}{16 b} + \frac{3 c d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac{5 c d x \cos ^{4}{\left (a + b x \right )}}{16 b} + \frac{3 d^{2} x^{2} \sin ^{4}{\left (a + b x \right )}}{32 b} + \frac{3 d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac{5 d^{2} x^{2} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac{3 c d \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{16 b^{2}} + \frac{5 c d \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{16 b^{2}} + \frac{3 d^{2} x \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{16 b^{2}} + \frac{5 d^{2} x \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{16 b^{2}} - \frac{d^{2} \sin ^{4}{\left (a + b x \right )}}{8 b^{3}} - \frac{5 d^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{3}} & \text{for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) \sin{\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 = 1.10493, size = 196, normalized size = 1.46 \begin{align*} -\frac{{\left (8 \, b^{2} d^{2} x^{2} + 16 \, b^{2} c d x + 8 \, b^{2} c^{2} - d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )}{256 \, b^{3}} - \frac{{\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{3}} + \frac{{\left (b d^{2} x + b c d\right )} \sin \left (4 \, b x + 4 \, a\right )}{64 \, b^{3}} + \frac{{\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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