Optimal. Leaf size=106 \[ -\frac{\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4} \]
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Rubi [A] time = 0.158925, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2862, 2669, 2635, 8} \[ -\frac{\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4} \]
Antiderivative was successfully verified.
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Rule 2862
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx &=-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac{1}{5} \int \cos ^2(c+d x) (2 b+2 a \sin (c+d x)) (a+b \sin (c+d x)) \, dx\\ &=-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac{1}{20} \int \cos ^2(c+d x) \left (10 a b+2 \left (a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac{\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac{1}{2} (a b) \int \cos ^2(c+d x) \, dx\\ &=-\frac{\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}+\frac{a b \cos (c+d x) \sin (c+d x)}{4 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac{1}{4} (a b) \int 1 \, dx\\ &=\frac{a b x}{4}-\frac{\left (a^2+4 b^2\right ) \cos ^3(c+d x)}{30 d}+\frac{a b \cos (c+d x) \sin (c+d x)}{4 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\\ \end{align*}
Mathematica [A] time = 0.365439, size = 77, normalized size = 0.73 \[ \frac{-30 \left (2 a^2+b^2\right ) \cos (c+d x)-5 \left (4 a^2+b^2\right ) \cos (3 (c+d x))+3 b (20 a (c+d x)-5 a \sin (4 (c+d x))+b \cos (5 (c+d x)))}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 94, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}}+2\,ab \left ( -1/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +1/8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/8\,dx+c/8 \right ) +{b}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{5}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{15}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15251, size = 92, normalized size = 0.87 \begin{align*} -\frac{80 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 16 \,{\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} b^{2}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38892, size = 185, normalized size = 1.75 \begin{align*} \frac{12 \, b^{2} \cos \left (d x + c\right )^{5} + 15 \, a b d x - 20 \,{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \,{\left (2 \, a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.36347, size = 172, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{a^{2} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac{a b x \sin ^{4}{\left (c + d x \right )}}{4} + \frac{a b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac{a b x \cos ^{4}{\left (c + d x \right )}}{4} + \frac{a b \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} - \frac{a b \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} - \frac{b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac{2 b^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{2} \sin{\left (c \right )} \cos ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20691, size = 111, normalized size = 1.05 \begin{align*} \frac{1}{4} \, a b x + \frac{b^{2} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac{a b \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac{{\left (4 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac{{\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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