Optimal. Leaf size=24 \[ -\frac {b \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d} \]
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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 = {2717, 2718}
\begin {gather*} \frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
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 = 0.02, size = 46, normalized size = 1.92 \begin {gather*} -\frac {b \cos (c) \cos (d x)}{d}+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d}+\frac {b \sin (c) \sin (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 25, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {-b \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\) | \(23\) |
default | \(-\frac {b \cos \left (d x +c \right )}{d}+\frac {a \sin \left (d x +c \right )}{d}\) | \(25\) |
risch | \(-\frac {b \cos \left (d x +c \right )}{d}+\frac {a \sin \left (d x +c \right )}{d}\) | \(25\) |
norman | \(\frac {\frac {2 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(50\) |
meijerg | \(\frac {\left (\cos \left (c \right ) \sqrt {\pi }\, a +\sqrt {\pi }\, \sin \left (c \right ) b \right ) \sin \left (d x \right )}{\sqrt {\pi }\, d}+\frac {\left (\cos \left (c \right ) \sqrt {\pi }\, b -\sqrt {\pi }\, \sin \left (c \right ) a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (d x \right )}{\sqrt {\pi }}\right )}{d}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 24, normalized size = 1.00 \begin {gather*} -\frac {b \cos \left (d x + c\right )}{d} + \frac {a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.85, size = 23, normalized size = 0.96 \begin {gather*} -\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 31, normalized size = 1.29 \begin {gather*} a \left (\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\left (c \right )} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} - \frac {\cos {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sin {\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 24, normalized size = 1.00 \begin {gather*} -\frac {b \cos \left (d x + c\right )}{d} + \frac {a \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 38, normalized size = 1.58 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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