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 = 0.57, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, 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 8
Rule 2633
Rule 2635
Rule 2748
Rule 2889
Rule 3023
Rule 3033
Rule 3049
Rule 3050
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*}
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Mathematica [A] time = 0.82, size = 157, normalized size = 0.68 \[ \frac {-35 \left (12 a^2 b+b^3\right ) \cos (3 (c+d x))+63 \left (4 a^2 b+b^3\right ) \cos (5 (c+d x))+105 a \left (-\left (2 a^2+3 b^2\right ) \sin (4 (c+d x))+8 a^2 c+8 a^2 d x-3 b^2 \sin (2 (c+d x))+b^2 \sin (6 (c+d x))+12 b^2 c+12 b^2 d x\right )-105 b \left (24 a^2+5 b^2\right ) \cos (c+d x)-15 b^3 \cos (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 141, normalized size = 0.61 \[ -\frac {240 \, b^{3} \cos \left (d x + c\right )^{7} - 336 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 560 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d x - 105 \, {\left (8 \, a b^{2} \cos \left (d x + c\right )^{5} - 2 \, {\left (2 \, a^{3} + 7 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 166, normalized size = 0.72 \[ -\frac {b^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a b^{2} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {3 \, a b^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {1}{16} \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x + \frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (24 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 196, normalized size = 0.84 \[ \frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a^{2} b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 131, normalized size = 0.56 \[ \frac {210 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 1344 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b - 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 64 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} b^{3}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.66, size = 455, normalized size = 1.96 \[ \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+3\,b^2\right )}{8\,\left (\frac {a^3}{4}+\frac {3\,a\,b^2}{8}\right )}\right )\,\left (2\,a^2+3\,b^2\right )}{8\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3}{4}+\frac {3\,a\,b^2}{8}\right )+\frac {4\,a^2\,b}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a\,b^2}{2}-a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {5\,a\,b^2}{2}-a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {a^3}{4}+\frac {3\,a\,b^2}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {11\,a^3}{4}+\frac {97\,a\,b^2}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {11\,a^3}{4}+\frac {97\,a\,b^2}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,a^2\,b-\frac {16\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {24\,a^2\,b}{5}+\frac {16\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (20\,a^2\,b+\frac {32\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {28\,a^2\,b}{5}+\frac {16\,b^3}{15}\right )+\frac {16\,b^3}{105}+12\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (2\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.29, size = 394, normalized size = 1.70 \[ \begin {cases} \frac {a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {2 a^{2} b \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {8 b^{3} \cos ^{7}{\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{3} \sin ^{2}{\relax (c )} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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