Optimal. Leaf size=301 \[ \frac{\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac{a \left (10 a^2-29 b^2\right ) \sin ^5(c+d x) \cos (c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}-\frac{\left (-44 a^2 b^2+15 a^4+6 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{630 b^2 d}+\frac{a \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{32 d}-\frac{3 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a b x}{64} \]
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Rubi [A] time = 0.653838, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2895, 3049, 3033, 3023, 2748, 2633, 2635, 8} \[ \frac{\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac{a \left (10 a^2-29 b^2\right ) \sin ^5(c+d x) \cos (c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}-\frac{\left (-44 a^2 b^2+15 a^4+6 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{630 b^2 d}+\frac{a \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{32 d}-\frac{3 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a b x}{64} \]
Antiderivative was successfully verified.
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Rule 2895
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{\int \sin ^3(c+d x) (a+b \sin (c+d x))^2 \left (24 \left (a^2-3 b^2\right )+2 a b \sin (c+d x)-10 \left (3 a^2-8 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{72 b^2}\\ &=-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{\int \sin ^3(c+d x) (a+b \sin (c+d x)) \left (8 a \left (6 a^2-23 b^2\right )+2 b \left (a^2-12 b^2\right ) \sin (c+d x)-6 a \left (10 a^2-29 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{504 b^2}\\ &=-\frac{a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{\int \sin ^3(c+d x) \left (48 a^2 \left (6 a^2-23 b^2\right )-378 a b^3 \sin (c+d x)-24 \left (15 a^4-44 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3024 b^2}\\ &=-\frac{\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac{a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{\int \sin ^3(c+d x) \left (-144 b^2 \left (9 a^2+4 b^2\right )-1890 a b^3 \sin (c+d x)\right ) \, dx}{15120 b^2}\\ &=-\frac{\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac{a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}+\frac{1}{8} (a b) \int \sin ^4(c+d x) \, dx-\frac{1}{105} \left (-9 a^2-4 b^2\right ) \int \sin ^3(c+d x) \, dx\\ &=-\frac{a b \cos (c+d x) \sin ^3(c+d x)}{32 d}-\frac{\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac{a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}+\frac{1}{32} (3 a b) \int \sin ^2(c+d x) \, dx-\frac{\left (9 a^2+4 b^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{105 d}\\ &=-\frac{\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac{\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{3 a b \cos (c+d x) \sin (c+d x)}{64 d}-\frac{a b \cos (c+d x) \sin ^3(c+d x)}{32 d}-\frac{\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac{a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}+\frac{1}{64} (3 a b) \int 1 \, dx\\ &=\frac{3 a b x}{64}-\frac{\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac{\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{3 a b \cos (c+d x) \sin (c+d x)}{64 d}-\frac{a b \cos (c+d x) \sin ^3(c+d x)}{32 d}-\frac{\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac{a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac{5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac{a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}\\ \end{align*}
Mathematica [A] time = 0.956657, size = 144, normalized size = 0.48 \[ \frac{-3780 \left (2 a^2+b^2\right ) \cos (c+d x)-840 \left (3 a^2+b^2\right ) \cos (3 (c+d x))+504 a^2 \cos (5 (c+d x))+360 a^2 \cos (7 (c+d x))-2520 a b \sin (4 (c+d x))+315 a b \sin (8 (c+d x))+7560 a b c+7560 a b d x+504 b^2 \cos (5 (c+d x))+90 b^2 \cos (7 (c+d x))-70 b^2 \cos (9 (c+d x))}{161280 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 161, normalized size = 0.5 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +2\,ab \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-1/16\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) }{64}}+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +{b}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{9}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{63}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998958, size = 135, normalized size = 0.45 \begin{align*} \frac{4608 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} + 315 \,{\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 512 \,{\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{2}}{161280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86421, size = 315, normalized size = 1.05 \begin{align*} -\frac{2240 \, b^{2} \cos \left (d x + c\right )^{9} - 2880 \,{\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{7} + 4032 \,{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} - 945 \, a b d x - 315 \,{\left (16 \, a b \cos \left (d x + c\right )^{7} - 24 \, a b \cos \left (d x + c\right )^{5} + 2 \, a b \cos \left (d x + c\right )^{3} + 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.89, size = 335, normalized size = 1.11 \begin{align*} \begin{cases} - \frac{a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{2 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac{3 a b x \sin ^{8}{\left (c + d x \right )}}{64} + \frac{3 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{9 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac{3 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac{3 a b x \cos ^{8}{\left (c + d x \right )}}{64} + \frac{3 a b \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{64 d} + \frac{11 a b \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} - \frac{11 a b \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac{3 a b \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac{b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac{8 b^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{4}{\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 = 1.35921, size = 192, normalized size = 0.64 \begin{align*} \frac{3}{64} \, a b x - \frac{b^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{a b \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac{a b \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (4 \, a^{2} + b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac{{\left (a^{2} + b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{{\left (3 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{3 \,{\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{128 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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