Optimal. Leaf size=28 \[ \frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 2637} \[ \frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rubi steps
\begin {align*} \int (c+d x) \sin (a+b x) \, dx &=-\frac {(c+d x) \cos (a+b x)}{b}+\frac {d \int \cos (a+b x) \, dx}{b}\\ &=-\frac {(c+d x) \cos (a+b x)}{b}+\frac {d \sin (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 27, normalized size = 0.96 \[ \frac {d \sin (a+b x)-b (c+d x) \cos (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 30, normalized size = 1.07 \[ -\frac {{\left (b d x + b c\right )} \cos \left (b x + a\right ) - d \sin \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.88, size = 31, normalized size = 1.11 \[ -\frac {{\left (b d x + b c\right )} \cos \left (b x + a\right )}{b^{2}} + \frac {d \sin \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 1.86 \[ \frac {\frac {d \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b}+\frac {d a \cos \left (b x +a \right )}{b}-c \cos \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 53, normalized size = 1.89 \[ -\frac {c \cos \left (b x + a\right ) - \frac {a d \cos \left (b x + a\right )}{b} + \frac {{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} d}{b}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 35, normalized size = 1.25 \[ \frac {d\,\sin \left (a+b\,x\right )}{b^2}-\frac {c\,\cos \left (a+b\,x\right )+d\,x\,\cos \left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 46, normalized size = 1.64 \[ \begin {cases} - \frac {c \cos {\left (a + b x \right )}}{b} - \frac {d x \cos {\left (a + b x \right )}}{b} + \frac {d \sin {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sin {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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