Optimal. Leaf size=120 \[ -\frac {3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b} \]
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Rubi [A] time = 0.08, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4404, 3311, 32, 2635, 8} \[ -\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}-\frac {3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b} \]
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 (a+b x) \, dx &=\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \sin ^2(a+b x) \, dx}{2 b}\\ &=\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \, dx}{4 b}+\frac {\left (3 d^3\right ) \int \sin ^2(a+b x) \, dx}{4 b^3}\\ &=-\frac {(c+d x)^3}{4 b}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {\left (3 d^3\right ) \int 1 \, dx}{8 b^3}\\ &=\frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 71, normalized size = 0.59 \[ \frac {3 d \sin (2 (a+b x)) \left (2 b^2 (c+d x)^2-d^2\right )-2 b (c+d x) \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-3 d^2\right )}{16 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 166, normalized size = 1.38 \[ \frac {2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} - 2 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x}{8 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 121, normalized size = 1.01 \[ -\frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 466, normalized size = 3.88 \[ \frac {\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {3 \left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{4}-\frac {3 \cos \left (b x +a \right ) \sin \left (b x +a \right )}{8}-\frac {3 b x}{8}-\frac {3 a}{8}-\frac {\left (b x +a \right )^{3}}{2}\right )}{b^{3}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{3}}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b}+\frac {a^{3} d^{3} \left (\cos ^{2}\left (b x +a \right )\right )}{2 b^{3}}-\frac {3 a^{2} c \,d^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{2 b^{2}}+\frac {3 a \,c^{2} d \left (\cos ^{2}\left (b x +a \right )\right )}{2 b}-\frac {c^{3} \left (\cos ^{2}\left (b x +a \right )\right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 342, normalized size = 2.85 \[ -\frac {8 \, c^{3} \cos \left (b x + a\right )^{2} - \frac {24 \, a c^{2} d \cos \left (b x + a\right )^{2}}{b} + \frac {24 \, a^{2} c d^{2} \cos \left (b x + a\right )^{2}}{b^{2}} - \frac {8 \, a^{3} d^{3} \cos \left (b x + a\right )^{2}}{b^{3}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} - \frac {12 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {6 \, {\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {6 \, {\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 165, normalized size = 1.38 \[ \frac {\cos \left (2\,a+2\,b\,x\right )\,\left (\frac {3\,c\,d^2}{4}-\frac {b^2\,c^3}{2}\right )}{2\,b^3}-\frac {3\,\sin \left (2\,a+2\,b\,x\right )\,\left (d^3-2\,b^2\,c^2\,d\right )}{16\,b^4}-\frac {d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )}{4\,b}+\frac {3\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )}{8\,b^2}+\frac {3\,x\,\cos \left (2\,a+2\,b\,x\right )\,\left (d^3-2\,b^2\,c^2\,d\right )}{8\,b^3}+\frac {3\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )}{4\,b^2}-\frac {3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.39, size = 342, normalized size = 2.85 \[ \begin {cases} - \frac {c^{3} \cos ^{2}{\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c^{2} d x \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {3 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c d^{2} x^{2} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {d^{3} x^{3} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {3 c^{2} d \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} + \frac {3 c d^{2} x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{2}} + \frac {3 d^{3} x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} + \frac {3 c d^{2} \cos ^{2}{\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} x \sin ^{2}{\left (a + b x \right )}}{8 b^{3}} + \frac {3 d^{3} x \cos ^{2}{\left (a + b x \right )}}{8 b^{3}} - \frac {3 d^{3} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin {\relax (a )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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