Optimal. Leaf size=354 \[ \frac{b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac{\left (-93 a^2 b^2+20 a^4+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac{5 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac{a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{a \left (-188 a^2 b^2+40 a^4+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}-\frac{a \left (8 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a x \left (8 a^2+9 b^2\right )+\frac{5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d} \]
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Rubi [A] time = 0.927725, antiderivative size = 354, 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, 2635, 8, 2633} \[ \frac{b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac{\left (-93 a^2 b^2+20 a^4+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac{5 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac{a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{a \left (-188 a^2 b^2+40 a^4+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}-\frac{a \left (8 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a x \left (8 a^2+9 b^2\right )+\frac{5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2895
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac{\int \sin ^2(c+d x) (a+b \sin (c+d x))^3 \left (3 \left (5 a^2-24 b^2\right )+3 a b \sin (c+d x)-20 \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{72 b^2}\\ &=-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac{\int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (3 a \left (15 a^2-88 b^2\right )+6 b \left (a^2-4 b^2\right ) \sin (c+d x)-3 a \left (20 a^2-87 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{504 b^2}\\ &=-\frac{a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac{\int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (9 a^2 \left (10 a^2-89 b^2\right )+3 a b \left (2 a^2-141 b^2\right ) \sin (c+d x)-6 \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3024 b^2}\\ &=-\frac{\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac{a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac{\int \sin ^2(c+d x) \left (45 a^3 \left (10 a^2-89 b^2\right )-144 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)-15 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{15120 b^2}\\ &=-\frac{a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac{\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac{a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac{\int \sin ^2(c+d x) \left (-945 a b^2 \left (8 a^2+9 b^2\right )-576 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx}{60480 b^2}\\ &=-\frac{a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac{\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac{a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}+\frac{1}{105} \left (b \left (27 a^2+4 b^2\right )\right ) \int \sin ^3(c+d x) \, dx+\frac{1}{64} \left (a \left (8 a^2+9 b^2\right )\right ) \int \sin ^2(c+d x) \, dx\\ &=-\frac{a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac{a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac{\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac{a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}+\frac{1}{128} \left (a \left (8 a^2+9 b^2\right )\right ) \int 1 \, dx-\frac{\left (b \left (27 a^2+4 b^2\right )\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{105 d}\\ &=\frac{1}{128} a \left (8 a^2+9 b^2\right ) x-\frac{b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac{b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac{a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac{a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac{\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac{a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac{5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}\\ \end{align*}
Mathematica [A] time = 1.21064, size = 204, normalized size = 0.58 \[ \frac{-3780 b \left (6 a^2+b^2\right ) \cos (c+d x)-840 \left (9 a^2 b+b^3\right ) \cos (3 (c+d x))+1512 a^2 b \cos (5 (c+d x))+1080 a^2 b \cos (7 (c+d x))+2520 a^3 \sin (2 (c+d x))-2520 a^3 \sin (4 (c+d x))-840 a^3 \sin (6 (c+d x))+10080 a^3 d x-3780 a b^2 \sin (4 (c+d x))+\frac{945}{2} a b^2 \sin (8 (c+d x))+15120 a b^2 c+11340 a b^2 d x+504 b^3 \cos (5 (c+d x))+90 b^3 \cos (7 (c+d x))-70 b^3 \cos (9 (c+d x))}{161280 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 218, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{\sin \left ( dx+c \right ) }{24} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{dx}{16}}+{\frac{c}{16}} \right ) +3\,{a}^{2}b \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +3\,a{b}^{2} \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}^{3} \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 = 1.12258, size = 189, normalized size = 0.53 \begin{align*} \frac{1680 \,{\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^{3} + 27648 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} b + 945 \,{\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^{2} - 1024 \,{\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^{3}}{322560 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94399, size = 405, normalized size = 1.14 \begin{align*} -\frac{4480 \, b^{3} \cos \left (d x + c\right )^{9} - 5760 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{7} + 8064 \,{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5} - 315 \,{\left (8 \, a^{3} + 9 \, a b^{2}\right )} d x - 105 \,{\left (144 \, a b^{2} \cos \left (d x + c\right )^{7} - 8 \,{\left (8 \, a^{3} + 27 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.0008, size = 505, normalized size = 1.43 \begin{align*} \begin{cases} \frac{a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{3 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{a^{3} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac{a^{3} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac{3 a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{6 a^{2} b \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac{9 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{9 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{27 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{9 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{9 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{33 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac{33 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac{9 a b^{2} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac{8 b^{3} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right )^{3} \sin ^{2}{\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 = 1.38893, size = 275, normalized size = 0.78 \begin{align*} -\frac{b^{3} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{3 \, a b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{1}{128} \,{\left (8 \, a^{3} + 9 \, a b^{2}\right )} x + \frac{{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac{{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{{\left (9 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{3 \,{\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac{{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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