Optimal. Leaf size=89 \[ -\frac {d^2 \sin ^2(a+b x)}{4 b^3}+\frac {d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac {c d x}{2 b}-\frac {d^2 x^2}{4 b} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4404, 3310} \[ \frac {d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}-\frac {d^2 \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac {c d x}{2 b}-\frac {d^2 x^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 4404
Rubi steps
\begin {align*} \int (c+d x)^2 \cos (a+b x) \sin (a+b x) \, dx &=\frac {(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac {d \int (c+d x) \sin ^2(a+b x) \, dx}{b}\\ &=\frac {d (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {d^2 \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac {d \int (c+d x) \, dx}{2 b}\\ &=-\frac {c d x}{2 b}-\frac {d^2 x^2}{4 b}+\frac {d (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {d^2 \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^2 \sin ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 50, normalized size = 0.56 \[ \frac {\cos (2 (a+b x)) \left (d^2-2 b^2 (c+d x)^2\right )+2 b d (c+d x) \sin (2 (a+b x))}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 92, normalized size = 1.03 \[ \frac {b^{2} d^{2} x^{2} + 2 \, b^{2} c d x - {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 73, normalized size = 0.82 \[ -\frac {{\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{3}} + \frac {{\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 215, normalized size = 2.42 \[ \frac {\frac {d^{2} \left (-\frac {\left (b x +a \right )^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{2}}-\frac {2 a \,d^{2} \left (-\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}+\frac {2 c d \left (-\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b}-\frac {a^{2} d^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{2 b^{2}}+\frac {a c d \left (\cos ^{2}\left (b x +a \right )\right )}{b}-\frac {c^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 171, normalized size = 1.92 \[ -\frac {4 \, c^{2} \cos \left (b x + a\right )^{2} - \frac {8 \, a c d \cos \left (b x + a\right )^{2}}{b} + \frac {4 \, a^{2} d^{2} \cos \left (b x + a\right )^{2}}{b^{2}} + \frac {2 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} c d}{b} - \frac {2 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{2}}{b^{2}}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 100, normalized size = 1.12 \[ \frac {\cos \left (2\,a+2\,b\,x\right )\,\left (\frac {d^2}{4}-\frac {b^2\,c^2}{2}\right )}{2\,b^3}+\frac {d^2\,x\,\sin \left (2\,a+2\,b\,x\right )}{4\,b^2}-\frac {d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )}{4\,b}+\frac {c\,d\,\sin \left (2\,a+2\,b\,x\right )}{4\,b^2}-\frac {c\,d\,x\,\cos \left (2\,a+2\,b\,x\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 175, normalized size = 1.97 \[ \begin {cases} - \frac {c^{2} \cos ^{2}{\left (a + b x \right )}}{2 b} + \frac {c d x \sin ^{2}{\left (a + b x \right )}}{2 b} - \frac {c d x \cos ^{2}{\left (a + b x \right )}}{2 b} + \frac {d^{2} x^{2} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {d^{2} x^{2} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {c d \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{2}} + \frac {d^{2} x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{2}} + \frac {d^{2} \cos ^{2}{\left (a + b x \right )}}{4 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin {\relax (a )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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