Optimal. Leaf size=196 \[ -\frac {3 d^3 \sin ^3(a+b x) \cos (a+b x)}{128 b^4}-\frac {45 d^3 \sin (a+b x) \cos (a+b x)}{256 b^4}-\frac {3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}-\frac {9 d^2 (c+d x) \sin ^2(a+b x)}{32 b^3}+\frac {3 d (c+d x)^2 \sin ^3(a+b x) \cos (a+b x)}{16 b^2}+\frac {9 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{32 b^2}+\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}+\frac {45 d^3 x}{256 b^3}-\frac {3 (c+d x)^3}{32 b} \]
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Rubi [A] time = 0.17, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4404, 3311, 32, 2635, 8} \[ -\frac {3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}-\frac {9 d^2 (c+d x) \sin ^2(a+b x)}{32 b^3}+\frac {3 d (c+d x)^2 \sin ^3(a+b x) \cos (a+b x)}{16 b^2}+\frac {9 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{32 b^2}-\frac {3 d^3 \sin ^3(a+b x) \cos (a+b x)}{128 b^4}-\frac {45 d^3 \sin (a+b x) \cos (a+b x)}{256 b^4}+\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}+\frac {45 d^3 x}{256 b^3}-\frac {3 (c+d x)^3}{32 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 3311
Rule 4404
Rubi steps
\begin {align*} \int (c+d x)^3 \cos (a+b x) \sin ^3(a+b x) \, dx &=\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}-\frac {(3 d) \int (c+d x)^2 \sin ^4(a+b x) \, dx}{4 b}\\ &=\frac {3 d (c+d x)^2 \cos (a+b x) \sin ^3(a+b x)}{16 b^2}-\frac {3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}+\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}-\frac {(9 d) \int (c+d x)^2 \sin ^2(a+b x) \, dx}{16 b}+\frac {\left (3 d^3\right ) \int \sin ^4(a+b x) \, dx}{32 b^3}\\ &=\frac {9 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{32 b^2}-\frac {9 d^2 (c+d x) \sin ^2(a+b x)}{32 b^3}-\frac {3 d^3 \cos (a+b x) \sin ^3(a+b x)}{128 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin ^3(a+b x)}{16 b^2}-\frac {3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}+\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}-\frac {(9 d) \int (c+d x)^2 \, dx}{32 b}+\frac {\left (9 d^3\right ) \int \sin ^2(a+b x) \, dx}{128 b^3}+\frac {\left (9 d^3\right ) \int \sin ^2(a+b x) \, dx}{32 b^3}\\ &=-\frac {3 (c+d x)^3}{32 b}-\frac {45 d^3 \cos (a+b x) \sin (a+b x)}{256 b^4}+\frac {9 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{32 b^2}-\frac {9 d^2 (c+d x) \sin ^2(a+b x)}{32 b^3}-\frac {3 d^3 \cos (a+b x) \sin ^3(a+b x)}{128 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin ^3(a+b x)}{16 b^2}-\frac {3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}+\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}+\frac {\left (9 d^3\right ) \int 1 \, dx}{256 b^3}+\frac {\left (9 d^3\right ) \int 1 \, dx}{64 b^3}\\ &=\frac {45 d^3 x}{256 b^3}-\frac {3 (c+d x)^3}{32 b}-\frac {45 d^3 \cos (a+b x) \sin (a+b x)}{256 b^4}+\frac {9 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{32 b^2}-\frac {9 d^2 (c+d x) \sin ^2(a+b x)}{32 b^3}-\frac {3 d^3 \cos (a+b x) \sin ^3(a+b x)}{128 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin ^3(a+b x)}{16 b^2}-\frac {3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}+\frac {(c+d x)^3 \sin ^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 135, normalized size = 0.69 \[ \frac {-64 b (c+d x) \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-3 d^2\right )+4 b (c+d x) \cos (4 (a+b x)) \left (8 b^2 (c+d x)^2-3 d^2\right )-6 d \sin (2 (a+b x)) \left (\cos (2 (a+b x)) \left (8 b^2 (c+d x)^2-d^2\right )-16 \left (2 b^2 (c+d x)^2-d^2\right )\right )}{1024 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 283, normalized size = 1.44 \[ \frac {40 \, b^{3} d^{3} x^{3} + 120 \, b^{3} c d^{2} x^{2} + 8 \, {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} - 8 \, {\left (16 \, b^{3} d^{3} x^{3} + 48 \, b^{3} c d^{2} x^{2} + 16 \, b^{3} c^{3} - 15 \, b c d^{2} + 3 \, {\left (16 \, b^{3} c^{2} d - 5 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (40 \, b^{3} c^{2} d - 17 \, b d^{3}\right )} x - 3 \, {\left (2 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} - {\left (40 \, b^{2} d^{3} x^{2} + 80 \, b^{2} c d^{2} x + 40 \, b^{2} c^{2} d - 17 \, d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{256 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.97, size = 241, normalized size = 1.23 \[ \frac {{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 24 \, b^{3} c^{2} d x + 8 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} - \frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} - \frac {3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{32 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 594, normalized size = 3.03 \[ \frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \left (\sin ^{4}\left (b x +a \right )\right )}{4}-\frac {3 \left (b x +a \right )^{2} \left (-\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{4}-\frac {3 \left (b x +a \right ) \left (\sin ^{4}\left (b x +a \right )\right )}{32}-\frac {3 \left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{128}-\frac {27 b x}{256}-\frac {27 a}{256}+\frac {9 \left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{32}-\frac {9 \cos \left (b x +a \right ) \sin \left (b x +a \right )}{64}+\frac {3 \left (b x +a \right )^{3}}{16}\right )}{b^{3}}-\frac {3 a \,d^{3} \left (\frac {\left (b x +a \right )^{2} \left (\sin ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \left (-\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}+\frac {3 \left (b x +a \right )^{2}}{32}-\frac {\left (\sin ^{4}\left (b x +a \right )\right )}{32}-\frac {3 \left (\sin ^{2}\left (b x +a \right )\right )}{32}\right )}{b^{3}}+\frac {3 c \,d^{2} \left (\frac {\left (b x +a \right )^{2} \left (\sin ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \left (-\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}+\frac {3 \left (b x +a \right )^{2}}{32}-\frac {\left (\sin ^{4}\left (b x +a \right )\right )}{32}-\frac {3 \left (\sin ^{2}\left (b x +a \right )\right )}{32}\right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (\frac {\left (b x +a \right ) \left (\sin ^{4}\left (b x +a \right )\right )}{4}+\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{16}-\frac {3 b x}{32}-\frac {3 a}{32}\right )}{b^{3}}-\frac {6 a c \,d^{2} \left (\frac {\left (b x +a \right ) \left (\sin ^{4}\left (b x +a \right )\right )}{4}+\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{16}-\frac {3 b x}{32}-\frac {3 a}{32}\right )}{b^{2}}+\frac {3 c^{2} d \left (\frac {\left (b x +a \right ) \left (\sin ^{4}\left (b x +a \right )\right )}{4}+\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{16}-\frac {3 b x}{32}-\frac {3 a}{32}\right )}{b}-\frac {a^{3} d^{3} \left (\sin ^{4}\left (b x +a \right )\right )}{4 b^{3}}+\frac {3 a^{2} c \,d^{2} \left (\sin ^{4}\left (b x +a \right )\right )}{4 b^{2}}-\frac {3 a \,c^{2} d \left (\sin ^{4}\left (b x +a \right )\right )}{4 b}+\frac {c^{3} \left (\sin ^{4}\left (b x +a \right )\right )}{4}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 549, normalized size = 2.80 \[ \frac {256 \, c^{3} \sin \left (b x + a\right )^{4} - \frac {768 \, a c^{2} d \sin \left (b x + a\right )^{4}}{b} + \frac {768 \, a^{2} c d^{2} \sin \left (b x + a\right )^{4}}{b^{2}} - \frac {256 \, a^{3} d^{3} \sin \left (b x + a\right )^{4}}{b^{3}} + \frac {24 \, {\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} - \frac {48 \, {\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {24 \, {\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {12 \, {\left ({\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 16 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {12 \, {\left ({\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 16 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 64 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + 96 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{1024 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 366, normalized size = 1.87 \[ -\frac {24\,d^3\,\sin \left (2\,a+2\,b\,x\right )-\frac {3\,d^3\,\sin \left (4\,a+4\,b\,x\right )}{4}+32\,b^3\,c^3\,\cos \left (2\,a+2\,b\,x\right )-8\,b^3\,c^3\,\cos \left (4\,a+4\,b\,x\right )-48\,b^2\,c^2\,d\,\sin \left (2\,a+2\,b\,x\right )+6\,b^2\,c^2\,d\,\sin \left (4\,a+4\,b\,x\right )+32\,b^3\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-8\,b^3\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-48\,b^2\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+6\,b^2\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )-48\,b\,c\,d^2\,\cos \left (2\,a+2\,b\,x\right )+3\,b\,c\,d^2\,\cos \left (4\,a+4\,b\,x\right )-48\,b\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+3\,b\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )+96\,b^3\,c^2\,d\,x\,\cos \left (2\,a+2\,b\,x\right )-24\,b^3\,c^2\,d\,x\,\cos \left (4\,a+4\,b\,x\right )-96\,b^2\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )+12\,b^2\,c\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )+96\,b^3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-24\,b^3\,c\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )}{256\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.28, size = 602, normalized size = 3.07 \[ \begin {cases} \frac {c^{3} \sin ^{4}{\left (a + b x \right )}}{4 b} + \frac {15 c^{2} d x \sin ^{4}{\left (a + b x \right )}}{32 b} - \frac {9 c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {9 c^{2} d x \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {15 c d^{2} x^{2} \sin ^{4}{\left (a + b x \right )}}{32 b} - \frac {9 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {9 c d^{2} x^{2} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {5 d^{3} x^{3} \sin ^{4}{\left (a + b x \right )}}{32 b} - \frac {3 d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {3 d^{3} x^{3} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {15 c^{2} d \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{32 b^{2}} + \frac {9 c^{2} d \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{32 b^{2}} + \frac {15 c d^{2} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{16 b^{2}} + \frac {9 c d^{2} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{16 b^{2}} + \frac {15 d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{32 b^{2}} + \frac {9 d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{32 b^{2}} - \frac {15 c d^{2} \sin ^{4}{\left (a + b x \right )}}{64 b^{3}} + \frac {9 c d^{2} \cos ^{4}{\left (a + b x \right )}}{64 b^{3}} - \frac {51 d^{3} x \sin ^{4}{\left (a + b x \right )}}{256 b^{3}} + \frac {9 d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{128 b^{3}} + \frac {45 d^{3} x \cos ^{4}{\left (a + b x \right )}}{256 b^{3}} - \frac {51 d^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{256 b^{4}} - \frac {45 d^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{3}{\relax (a )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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