Optimal. Leaf size=260 \[ -\frac {3 d^4 \cos ^4(a+b x)}{128 b^5}-\frac {45 d^4 \cos ^2(a+b x)}{128 b^5}-\frac {3 d^3 (c+d x) \sin (a+b x) \cos ^3(a+b x)}{32 b^4}-\frac {45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}+\frac {3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}+\frac {9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}+\frac {d (c+d x)^3 \sin (a+b x) \cos ^3(a+b x)}{4 b^2}+\frac {3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac {45 c d^3 x}{64 b^3}-\frac {45 d^4 x^2}{128 b^3}+\frac {3 (c+d x)^4}{32 b} \]
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Rubi [A] time = 0.23, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4405, 3311, 32, 3310} \[ \frac {3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}+\frac {9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \sin (a+b x) \cos ^3(a+b x)}{32 b^4}-\frac {45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}+\frac {d (c+d x)^3 \sin (a+b x) \cos ^3(a+b x)}{4 b^2}+\frac {3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}-\frac {3 d^4 \cos ^4(a+b x)}{128 b^5}-\frac {45 d^4 \cos ^2(a+b x)}{128 b^5}-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac {45 c d^3 x}{64 b^3}-\frac {45 d^4 x^2}{128 b^3}+\frac {3 (c+d x)^4}{32 b} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3310
Rule 3311
Rule 4405
Rubi steps
\begin {align*} \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx &=-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}+\frac {d \int (c+d x)^3 \cos ^4(a+b x) \, dx}{b}\\ &=\frac {3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}+\frac {d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}+\frac {(3 d) \int (c+d x)^3 \cos ^2(a+b x) \, dx}{4 b}-\frac {\left (3 d^3\right ) \int (c+d x) \cos ^4(a+b x) \, dx}{8 b^3}\\ &=\frac {9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac {3 d^4 \cos ^4(a+b x)}{128 b^5}+\frac {3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac {3 d^3 (c+d x) \cos ^3(a+b x) \sin (a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}+\frac {(3 d) \int (c+d x)^3 \, dx}{8 b}-\frac {\left (9 d^3\right ) \int (c+d x) \cos ^2(a+b x) \, dx}{32 b^3}-\frac {\left (9 d^3\right ) \int (c+d x) \cos ^2(a+b x) \, dx}{8 b^3}\\ &=\frac {3 (c+d x)^4}{32 b}-\frac {45 d^4 \cos ^2(a+b x)}{128 b^5}+\frac {9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac {3 d^4 \cos ^4(a+b x)}{128 b^5}+\frac {3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac {45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac {3 d^3 (c+d x) \cos ^3(a+b x) \sin (a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}-\frac {\left (9 d^3\right ) \int (c+d x) \, dx}{64 b^3}-\frac {\left (9 d^3\right ) \int (c+d x) \, dx}{16 b^3}\\ &=-\frac {45 c d^3 x}{64 b^3}-\frac {45 d^4 x^2}{128 b^3}+\frac {3 (c+d x)^4}{32 b}-\frac {45 d^4 \cos ^2(a+b x)}{128 b^5}+\frac {9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac {3 d^4 \cos ^4(a+b x)}{128 b^5}+\frac {3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac {(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac {45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac {3 d^3 (c+d x) \cos ^3(a+b x) \sin (a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 1.85, size = 158, normalized size = 0.61 \[ -\frac {-8 b d (c+d x) \sin (2 (a+b x)) \left (\cos (2 (a+b x)) \left (8 b^2 (c+d x)^2-3 d^2\right )+16 \left (2 b^2 (c+d x)^2-3 d^2\right )\right )+64 \cos (2 (a+b x)) \left (2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right )+\cos (4 (a+b x)) \left (32 b^4 (c+d x)^4-24 b^2 d^2 (c+d x)^2+3 d^4\right )}{1024 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 378, normalized size = 1.45 \[ \frac {12 \, b^{4} d^{4} x^{4} + 48 \, b^{4} c d^{3} x^{3} - {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 9 \, {\left (8 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 9 \, {\left (8 \, b^{2} d^{4} x^{2} + 16 \, b^{2} c d^{3} x + 8 \, b^{2} c^{2} d^{2} - 5 \, d^{4}\right )} \cos \left (b x + a\right )^{2} + 6 \, {\left (8 \, b^{4} c^{3} d - 15 \, b^{2} c d^{3}\right )} x + 2 \, {\left (2 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 15 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - 5 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 361, normalized size = 1.39 \[ -\frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{5}} + \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} + \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 1150, normalized size = 4.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 967, normalized size = 3.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 576, normalized size = 2.22 \[ -\frac {192\,d^4\,\cos \left (2\,a+2\,b\,x\right )+3\,d^4\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )+32\,b^4\,c^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )-32\,b^3\,c^3\,d\,\sin \left (4\,a+4\,b\,x\right )-384\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )-24\,b^2\,c^2\,d^2\,\cos \left (4\,a+4\,b\,x\right )-384\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )-24\,b^2\,d^4\,x^2\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )+32\,b^4\,d^4\,x^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )-32\,b^3\,d^4\,x^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )+12\,b\,c\,d^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )+12\,b\,d^4\,x\,\sin \left (4\,a+4\,b\,x\right )+768\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )+192\,b^4\,c^2\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )-768\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+512\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )-48\,b^2\,c\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^3\,d\,x\,\cos \left (4\,a+4\,b\,x\right )+512\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )+128\,b^4\,c\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-768\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-768\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )-96\,b^3\,c^2\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )-96\,b^3\,c\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{1024\,b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.23, size = 935, normalized size = 3.60 \[ \begin {cases} - \frac {c^{4} \cos ^{4}{\left (a + b x \right )}}{4 b} + \frac {3 c^{3} d x \sin ^{4}{\left (a + b x \right )}}{8 b} + \frac {3 c^{3} d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} - \frac {5 c^{3} d x \cos ^{4}{\left (a + b x \right )}}{8 b} + \frac {9 c^{2} d^{2} x^{2} \sin ^{4}{\left (a + b x \right )}}{16 b} + \frac {9 c^{2} d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {15 c^{2} d^{2} x^{2} \cos ^{4}{\left (a + b x \right )}}{16 b} + \frac {3 c d^{3} x^{3} \sin ^{4}{\left (a + b x \right )}}{8 b} + \frac {3 c d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} - \frac {5 c d^{3} x^{3} \cos ^{4}{\left (a + b x \right )}}{8 b} + \frac {3 d^{4} x^{4} \sin ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 d^{4} x^{4} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {5 d^{4} x^{4} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 c^{3} d \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {5 c^{3} d \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} + \frac {9 c^{2} d^{2} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {15 c^{2} d^{2} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} + \frac {9 c d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {15 c d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} + \frac {3 d^{4} x^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {5 d^{4} x^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} - \frac {9 c^{2} d^{2} \sin ^{4}{\left (a + b x \right )}}{32 b^{3}} + \frac {15 c^{2} d^{2} \cos ^{4}{\left (a + b x \right )}}{32 b^{3}} - \frac {45 c d^{3} x \sin ^{4}{\left (a + b x \right )}}{64 b^{3}} - \frac {9 c d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{3}} + \frac {51 c d^{3} x \cos ^{4}{\left (a + b x \right )}}{64 b^{3}} - \frac {45 d^{4} x^{2} \sin ^{4}{\left (a + b x \right )}}{128 b^{3}} - \frac {9 d^{4} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{64 b^{3}} + \frac {51 d^{4} x^{2} \cos ^{4}{\left (a + b x \right )}}{128 b^{3}} - \frac {45 c d^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{4}} - \frac {51 c d^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{4}} - \frac {45 d^{4} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{4}} - \frac {51 d^{4} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{4}} + \frac {45 d^{4} \sin ^{4}{\left (a + b x \right )}}{256 b^{5}} - \frac {51 d^{4} \cos ^{4}{\left (a + b x \right )}}{256 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sin {\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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