Optimal. Leaf size=75 \[ \frac{d \sin ^3(a+b x)}{9 b^2}+\frac{2 d \sin (a+b x)}{3 b^2}-\frac{2 (c+d x) \cos (a+b x)}{3 b}-\frac{(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
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Rubi [A] time = 0.0418427, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3310, 3296, 2637} \[ \frac{d \sin ^3(a+b x)}{9 b^2}+\frac{2 d \sin (a+b x)}{3 b^2}-\frac{2 (c+d x) \cos (a+b x)}{3 b}-\frac{(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x) \sin ^3(a+b x) \, dx &=-\frac{(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{d \sin ^3(a+b x)}{9 b^2}+\frac{2}{3} \int (c+d x) \sin (a+b x) \, dx\\ &=-\frac{2 (c+d x) \cos (a+b x)}{3 b}-\frac{(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{d \sin ^3(a+b x)}{9 b^2}+\frac{(2 d) \int \cos (a+b x) \, dx}{3 b}\\ &=-\frac{2 (c+d x) \cos (a+b x)}{3 b}+\frac{2 d \sin (a+b x)}{3 b^2}-\frac{(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{d \sin ^3(a+b x)}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.172696, size = 59, normalized size = 0.79 \[ \frac{-27 b (c+d x) \cos (a+b x)+3 b (c+d x) \cos (3 (a+b x))+d (27 \sin (a+b x)-\sin (3 (a+b x)))}{36 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 95, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ( -{\frac{ \left ( bx+a \right ) \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{9}}+{\frac{2\,\sin \left ( bx+a \right ) }{3}} \right ) }+{\frac{da \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3\,b}}-{\frac{c \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05489, size = 140, normalized size = 1.87 \begin{align*} \frac{12 \,{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} c - \frac{12 \,{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} a d}{b} + \frac{{\left (3 \,{\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 27 \,{\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) + 27 \, \sin \left (b x + a\right )\right )} d}{b}}{36 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63869, size = 153, normalized size = 2.04 \begin{align*} \frac{3 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - 9 \,{\left (b d x + b c\right )} \cos \left (b x + a\right ) -{\left (d \cos \left (b x + a\right )^{2} - 7 \, d\right )} \sin \left (b x + a\right )}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.11446, size = 126, normalized size = 1.68 \begin{align*} \begin{cases} - \frac{c \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{2 c \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac{d x \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{2 d x \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac{7 d \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{2 d \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin ^{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 = 1.15119, size = 93, normalized size = 1.24 \begin{align*} \frac{{\left (b d x + b c\right )} \cos \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} - \frac{3 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )}{4 \, b^{2}} - \frac{d \sin \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac{3 \, d \sin \left (b x + a\right )}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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