Optimal. Leaf size=75 \[ \frac{d \cos ^3(a+b x)}{9 b^2}+\frac{2 d \cos (a+b x)}{3 b^2}+\frac{2 (c+d x) \sin (a+b x)}{3 b}+\frac{(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.0417872, 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, 2638} \[ \frac{d \cos ^3(a+b x)}{9 b^2}+\frac{2 d \cos (a+b x)}{3 b^2}+\frac{2 (c+d x) \sin (a+b x)}{3 b}+\frac{(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x) \cos ^3(a+b x) \, dx &=\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac{2}{3} \int (c+d x) \cos (a+b x) \, dx\\ &=\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{2 (c+d x) \sin (a+b x)}{3 b}+\frac{(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac{(2 d) \int \sin (a+b x) \, dx}{3 b}\\ &=\frac{2 d \cos (a+b x)}{3 b^2}+\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{2 (c+d x) \sin (a+b x)}{3 b}+\frac{(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.164027, size = 52, normalized size = 0.69 \[ \frac{3 b (c+d x) (9 \sin (a+b x)+\sin (3 (a+b x)))+27 d \cos (a+b x)+d \cos (3 (a+b x))}{36 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 95, normalized size = 1.3 \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}}{9}}+{\frac{2\,\cos \left ( bx+a \right ) }{3}} \right ) }-{\frac{da \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3\,b}}+{\frac{c \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \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 = 0.996978, size = 139, normalized size = 1.85 \begin{align*} -\frac{12 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} c - \frac{12 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} a d}{b} - \frac{{\left (3 \,{\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) + 27 \,{\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) + 27 \, \cos \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 = 1.37993, size = 153, normalized size = 2.04 \begin{align*} \frac{d \cos \left (b x + a\right )^{3} + 6 \, d \cos \left (b x + a\right ) + 3 \,{\left (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 )}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18023, size = 126, normalized size = 1.68 \begin{align*} \begin{cases} \frac{2 c \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{c \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac{2 d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{d x \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac{2 d \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{3 b^{2}} + \frac{7 d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\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 = 1.19975, size = 93, normalized size = 1.24 \begin{align*} \frac{d \cos \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac{3 \, d \cos \left (b x + a\right )}{4 \, b^{2}} + \frac{{\left (b d x + b c\right )} \sin \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} + \frac{3 \,{\left (b d x + b c\right )} \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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