Optimal. Leaf size=194 \[ -\frac {a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}+\frac {b \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 b \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3}{128} b x \left (8 a^2+b^2\right )-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d} \]
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Rubi [A] time = 0.33, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2862, 2669, 2635, 8} \[ -\frac {a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}+\frac {b \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 b \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3}{128} b x \left (8 a^2+b^2\right )-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2862
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx &=-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac {1}{8} \int \cos ^4(c+d x) (3 b+3 a \sin (c+d x)) (a+b \sin (c+d x))^2 \, dx\\ &=-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac {1}{56} \int \cos ^4(c+d x) (a+b \sin (c+d x)) \left (27 a b+3 \left (2 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac {1}{336} \int \cos ^4(c+d x) \left (21 b \left (8 a^2+b^2\right )+3 a \left (2 a^2+61 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac {1}{16} \left (b \left (8 a^2+b^2\right )\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}+\frac {b \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac {1}{64} \left (3 b \left (8 a^2+b^2\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}+\frac {3 b \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {b \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}+\frac {1}{128} \left (3 b \left (8 a^2+b^2\right )\right ) \int 1 \, dx\\ &=\frac {3}{128} b \left (8 a^2+b^2\right ) x-\frac {a \left (2 a^2+61 b^2\right ) \cos ^5(c+d x)}{560 d}+\frac {3 b \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {b \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {\left (2 a^2+7 b^2\right ) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}-\frac {3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 189, normalized size = 0.97 \[ \frac {-280 \left (4 a^3+3 a b^2\right ) \cos (3 (c+d x))-224 a^3 \cos (5 (c+d x))-280 a \left (8 a^2+9 b^2\right ) \cos (c+d x)+840 a^2 b \sin (2 (c+d x))-840 a^2 b \sin (4 (c+d x))-280 a^2 b \sin (6 (c+d x))+3360 a^2 b c+3360 a^2 b d x+168 a b^2 \cos (5 (c+d x))+120 a b^2 \cos (7 (c+d x))-140 b^3 \sin (4 (c+d x))+\frac {35}{2} b^3 \sin (8 (c+d x))+840 b^3 c+420 b^3 d x}{17920 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 136, normalized size = 0.70 \[ \frac {1920 \, a b^{2} \cos \left (d x + c\right )^{7} - 896 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (8 \, a^{2} b + b^{3}\right )} d x + 35 \, {\left (16 \, b^{3} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} b + 3 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (8 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 184, normalized size = 0.95 \[ \frac {3 \, a b^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {b^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{2} b \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} + \frac {3 \, a^{2} b \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {3}{128} \, {\left (8 \, a^{2} b + b^{3}\right )} x - \frac {{\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (4 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac {{\left (6 \, a^{2} b + b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 180, normalized size = 0.93 \[ \frac {-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+b^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 117, normalized size = 0.60 \[ -\frac {7168 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b - 3072 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a b^{2} - 35 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{3}}{35840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.92, size = 552, normalized size = 2.85 \[ \frac {3\,b\,\mathrm {atan}\left (\frac {3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^2+b^2\right )}{64\,\left (\frac {3\,a^2\,b}{8}+\frac {3\,b^3}{64}\right )}\right )\,\left (8\,a^2+b^2\right )}{64\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2\,b}{8}+\frac {3\,b^3}{64}\right )+10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {12\,a\,b^2}{35}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (6\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (14\,a^3+12\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {12\,a\,b^2}{5}-\frac {26\,a^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {6\,a^3}{5}+\frac {96\,a\,b^2}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {62\,a^3}{5}+\frac {96\,a\,b^2}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {3\,a^2\,b}{8}+\frac {3\,b^3}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {41\,a^2\,b}{8}-\frac {23\,b^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {41\,a^2\,b}{8}-\frac {23\,b^3}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {13\,a^2\,b}{8}+\frac {333\,b^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {13\,a^2\,b}{8}+\frac {333\,b^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {31\,a^2\,b}{8}+\frac {671\,b^3}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {31\,a^2\,b}{8}+\frac {671\,b^3}{64}\right )+\frac {2\,a^3}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {3\,b\,\left (8\,a^2+b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.06, size = 456, normalized size = 2.35 \[ \begin {cases} - \frac {a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a^{2} b x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{2} b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{2} b x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a^{2} b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a^{2} b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {6 a b^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {3 b^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 b^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 b^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {11 b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 b^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{3} \sin {\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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