Optimal. Leaf size=72 \[ \frac{d \sin ^3(a+b x) \cos (a+b x)}{16 b^2}+\frac{3 d \sin (a+b x) \cos (a+b x)}{32 b^2}+\frac{(c+d x) \sin ^4(a+b x)}{4 b}-\frac{3 d x}{32 b} \]
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Rubi [A] time = 0.04547, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4404, 2635, 8} \[ \frac{d \sin ^3(a+b x) \cos (a+b x)}{16 b^2}+\frac{3 d \sin (a+b x) \cos (a+b x)}{32 b^2}+\frac{(c+d x) \sin ^4(a+b x)}{4 b}-\frac{3 d x}{32 b} \]
Antiderivative was successfully verified.
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Rule 4404
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (c+d x) \cos (a+b x) \sin ^3(a+b x) \, dx &=\frac{(c+d x) \sin ^4(a+b x)}{4 b}-\frac{d \int \sin ^4(a+b x) \, dx}{4 b}\\ &=\frac{d \cos (a+b x) \sin ^3(a+b x)}{16 b^2}+\frac{(c+d x) \sin ^4(a+b x)}{4 b}-\frac{(3 d) \int \sin ^2(a+b x) \, dx}{16 b}\\ &=\frac{3 d \cos (a+b x) \sin (a+b x)}{32 b^2}+\frac{d \cos (a+b x) \sin ^3(a+b x)}{16 b^2}+\frac{(c+d x) \sin ^4(a+b x)}{4 b}-\frac{(3 d) \int 1 \, dx}{32 b}\\ &=-\frac{3 d x}{32 b}+\frac{3 d \cos (a+b x) \sin (a+b x)}{32 b^2}+\frac{d \cos (a+b x) \sin ^3(a+b x)}{16 b^2}+\frac{(c+d x) \sin ^4(a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.116165, size = 75, normalized size = 1.04 \[ \frac{d (\sin (2 (a+b x))-2 b x \cos (2 (a+b x)))}{16 b^2}-\frac{d (\sin (4 (a+b x))-4 b x \cos (4 (a+b x)))}{128 b^2}+\frac{c \sin ^4(a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 85, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ({\frac{ \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{4}}+{\frac{\cos \left ( bx+a \right ) }{16} \left ( \left ( \sin \left ( bx+a \right ) \right ) ^{3}+{\frac{3\,\sin \left ( bx+a \right ) }{2}} \right ) }-{\frac{3\,bx}{32}}-{\frac{3\,a}{32}} \right ) }-{\frac{ad \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{4\,b}}+{\frac{c \left ( \sin \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 [A] time = 1.09336, size = 124, normalized size = 1.72 \begin{align*} \frac{32 \, c \sin \left (b x + a\right )^{4} - \frac{32 \, a d \sin \left (b x + a\right )^{4}}{b} + \frac{{\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 )} d}{b}}{128 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.473922, size = 192, normalized size = 2.67 \begin{align*} \frac{8 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{4} + 5 \, b d x - 16 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} -{\left (2 \, d \cos \left (b x + a\right )^{3} - 5 \, d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{32 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.37126, size = 160, normalized size = 2.22 \begin{align*} \begin{cases} - \frac{c \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{2 b} - \frac{c \cos ^{4}{\left (a + b x \right )}}{4 b} + \frac{5 d x \sin ^{4}{\left (a + b x \right )}}{32 b} - \frac{3 d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac{3 d x \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac{5 d \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{32 b^{2}} + \frac{3 d \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{32 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin ^{3}{\left (a \right )} \cos{\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 = 1.11426, size = 101, normalized size = 1.4 \begin{align*} \frac{{\left (b d x + b c\right )} \cos \left (4 \, b x + 4 \, a\right )}{32 \, b^{2}} - \frac{{\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{2}} - \frac{d \sin \left (4 \, b x + 4 \, a\right )}{128 \, b^{2}} + \frac{d \sin \left (2 \, b x + 2 \, a\right )}{16 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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