Optimal. Leaf size=24 \[ \frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2637, 2638} \[ \frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin {align*} \int (a \cos (c+d x)+b \sin (c+d x)) \, dx &=a \int \cos (c+d x) \, dx+b \int \sin (c+d x) \, dx\\ &=-\frac {b \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.92 \[ \frac {a \sin (c) \cos (d x)}{d}+\frac {a \cos (c) \sin (d x)}{d}+\frac {b \sin (c) \sin (d x)}{d}-\frac {b \cos (c) \cos (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 23, normalized size = 0.96 \[ -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 24, normalized size = 1.00 \[ -\frac {b \cos \left (d x + c\right )}{d} + \frac {a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.04 \[ -\frac {b \cos \left (d x +c \right )}{d}+\frac {a \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 24, normalized size = 1.00 \[ -\frac {b \cos \left (d x + c\right )}{d} + \frac {a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 38, normalized size = 1.58 \[ -\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 31, normalized size = 1.29 \[ a \left (\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\relax (c )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} - \frac {\cos {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sin {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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