Optimal. Leaf size=232 \[ \frac{b \left (21 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac{b \left (21 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac{b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac{a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{56 d}-\frac{a \left (2 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left (2 a^2+3 b^2\right )+\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{14 d} \]
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Rubi [A] time = 0.571816, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2889, 3050, 3049, 3033, 3023, 2748, 2635, 8, 2633} \[ \frac{b \left (21 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac{b \left (21 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac{b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac{a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{56 d}-\frac{a \left (2 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left (2 a^2+3 b^2\right )+\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{14 d} \]
Antiderivative was successfully verified.
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Rule 2889
Rule 3050
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \sin ^2(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx\\ &=\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{1}{7} \int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (4 a+b \sin (c+d x)-3 a \sin ^2(c+d x)\right ) \, dx\\ &=\frac{a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{1}{42} \int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (15 a^2+15 a b \sin (c+d x)-6 \left (a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx\\ &=\frac{b \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac{a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{1}{210} \int \sin ^2(c+d x) \left (75 a^3+6 b \left (21 a^2+4 b^2\right ) \sin (c+d x)-15 a \left (2 a^2-7 b^2\right ) \sin ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (2 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{56 d}+\frac{b \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac{a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{1}{840} \int \sin ^2(c+d x) \left (105 a \left (2 a^2+3 b^2\right )+24 b \left (21 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac{a \left (2 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{56 d}+\frac{b \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac{a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{1}{8} \left (a \left (2 a^2+3 b^2\right )\right ) \int \sin ^2(c+d x) \, dx+\frac{1}{35} \left (b \left (21 a^2+4 b^2\right )\right ) \int \sin ^3(c+d x) \, dx\\ &=-\frac{a \left (2 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (2 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{56 d}+\frac{b \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac{a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{1}{16} \left (a \left (2 a^2+3 b^2\right )\right ) \int 1 \, dx-\frac{\left (b \left (21 a^2+4 b^2\right )\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{35 d}\\ &=\frac{1}{16} a \left (2 a^2+3 b^2\right ) x-\frac{b \left (21 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac{b \left (21 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac{a \left (2 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (2 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{56 d}+\frac{b \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac{a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac{\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d}\\ \end{align*}
Mathematica [A] time = 0.808312, size = 157, normalized size = 0.68 \[ \frac{105 a \left (-\left (2 a^2+3 b^2\right ) \sin (4 (c+d x))+8 a^2 c+8 a^2 d x-3 b^2 \sin (2 (c+d x))+b^2 \sin (6 (c+d x))+12 b^2 c+12 b^2 d x\right )-105 b \left (24 a^2+5 b^2\right ) \cos (c+d x)-35 \left (12 a^2 b+b^3\right ) \cos (3 (c+d x))+63 \left (4 a^2 b+b^3\right ) \cos (5 (c+d x))-15 b^3 \cos (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 196, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4}}+{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8}}+{\frac{dx}{8}}+{\frac{c}{8}} \right ) +3\,{a}^{2}b \left ( -1/5\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-2/15\, \left ( \cos \left ( dx+c \right ) \right ) ^{3} \right ) +3\,a{b}^{2} \left ( -1/6\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-1/8\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +1/16\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) +{b}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{7}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{35}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{105}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1982, size = 177, normalized size = 0.76 \begin{align*} \frac{210 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 1344 \,{\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b - 105 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 64 \,{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} b^{3}}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5761, size = 343, normalized size = 1.48 \begin{align*} -\frac{240 \, b^{3} \cos \left (d x + c\right )^{7} - 336 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 560 \,{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} - 105 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} d x - 105 \,{\left (8 \, a b^{2} \cos \left (d x + c\right )^{5} - 2 \,{\left (2 \, a^{3} + 7 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.26901, size = 394, normalized size = 1.7 \begin{align*} \begin{cases} \frac{a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} - \frac{a^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac{a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac{2 a^{2} b \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac{3 a b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{9 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{3 a b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{3 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} - \frac{a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac{3 a b^{2} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac{b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac{8 b^{3} \cos ^{7}{\left (c + d x \right )}}{105 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{3} \sin ^{2}{\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 = 1.18634, size = 224, normalized size = 0.97 \begin{align*} -\frac{b^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{a b^{2} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac{3 \, a b^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{1}{16} \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} x + \frac{3 \,{\left (4 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{{\left (24 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac{{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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