Optimal. Leaf size=27 \[ \frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, 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 = {3377, 2718}
\begin {gather*} \frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rubi steps
\begin {align*} \int (c+d x) \cos (a+b x) \, dx &=\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x) \, dx}{b}\\ &=\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 0.96 \begin {gather*} \frac {d \cos (a+b x)+b (c+d x) \sin (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 51, normalized size = 1.89
method | result | size |
risch | \(\frac {d \cos \left (b x +a \right )}{b^{2}}+\frac {\left (d x +c \right ) \sin \left (b x +a \right )}{b}\) | \(28\) |
derivativedivides | \(\frac {-\frac {d a \sin \left (b x +a \right )}{b}+c \sin \left (b x +a \right )+\frac {d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}}{b}\) | \(51\) |
default | \(\frac {-\frac {d a \sin \left (b x +a \right )}{b}+c \sin \left (b x +a \right )+\frac {d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}}{b}\) | \(51\) |
norman | \(\frac {\frac {2 d}{b^{2}}+\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {2 d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(55\) |
meijerg | \(\frac {2 d \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {2 d \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x b \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {c \cos \left (a \right ) \sin \left (b x \right )}{b}-\frac {c \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 50, normalized size = 1.85 \begin {gather*} \frac {c \sin \left (b x + a\right ) - \frac {a d \sin \left (b x + a\right )}{b} + \frac {{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} d}{b}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 28, normalized size = 1.04 \begin {gather*} \frac {d \cos \left (b x + a\right ) + {\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 46, normalized size = 1.70 \begin {gather*} \begin {cases} \frac {c \sin {\left (a + b x \right )}}{b} + \frac {d x \sin {\left (a + b x \right )}}{b} + \frac {d \cos {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cos {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 30, normalized size = 1.11 \begin {gather*} \frac {d \cos \left (b x + a\right )}{b^{2}} + \frac {{\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 34, normalized size = 1.26 \begin {gather*} \frac {c\,\sin \left (a+b\,x\right )+d\,x\,\sin \left (a+b\,x\right )}{b}+\frac {d\,\cos \left (a+b\,x\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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