Optimal. Leaf size=37 \[ -\frac {b e^x \cos (a+b x)}{1+b^2}+\frac {e^x \sin (a+b x)}{1+b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4517}
\begin {gather*} \frac {e^x \sin (a+b x)}{b^2+1}-\frac {b e^x \cos (a+b x)}{b^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 4517
Rubi steps
\begin {align*} \int e^x \sin (a+b x) \, dx &=-\frac {b e^x \cos (a+b x)}{1+b^2}+\frac {e^x \sin (a+b x)}{1+b^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 27, normalized size = 0.73 \begin {gather*} \frac {e^x (-b \cos (a+b x)+\sin (a+b x))}{1+b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 36, normalized size = 0.97
method | result | size |
default | \(-\frac {b \,{\mathrm e}^{x} \cos \left (b x +a \right )}{b^{2}+1}+\frac {{\mathrm e}^{x} \sin \left (b x +a \right )}{b^{2}+1}\) | \(36\) |
risch | \(-\frac {{\mathrm e}^{x} {\mathrm e}^{i b x} {\mathrm e}^{i a}}{2 \left (b -i\right )}-\frac {{\mathrm e}^{x} {\mathrm e}^{-i b x} {\mathrm e}^{-i a}}{2 \left (i+b \right )}\) | \(44\) |
norman | \(\frac {\frac {b \,{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}+1}-\frac {b \,{\mathrm e}^{x}}{b^{2}+1}+\frac {2 \,{\mathrm e}^{x} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}+1}}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 28, normalized size = 0.76 \begin {gather*} -\frac {{\left (b \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} e^{x}}{b^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.00, size = 30, normalized size = 0.81 \begin {gather*} -\frac {b \cos \left (b x + a\right ) e^{x} - e^{x} \sin \left (b x + a\right )}{b^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.25, size = 116, normalized size = 3.14 \begin {gather*} \begin {cases} \frac {x e^{x} \sin {\left (a - i x \right )}}{2} + \frac {i x e^{x} \cos {\left (a - i x \right )}}{2} - \frac {i e^{x} \cos {\left (a - i x \right )}}{2} & \text {for}\: b = - i \\\frac {x e^{x} \sin {\left (a + i x \right )}}{2} - \frac {i x e^{x} \cos {\left (a + i x \right )}}{2} + \frac {i e^{x} \cos {\left (a + i x \right )}}{2} & \text {for}\: b = i \\- \frac {b e^{x} \cos {\left (a + b x \right )}}{b^{2} + 1} + \frac {e^{x} \sin {\left (a + b x \right )}}{b^{2} + 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 35, normalized size = 0.95 \begin {gather*} -{\left (\frac {b \cos \left (b x + a\right )}{b^{2} + 1} - \frac {\sin \left (b x + a\right )}{b^{2} + 1}\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 26, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^x\,\left (\sin \left (a+b\,x\right )-b\,\cos \left (a+b\,x\right )\right )}{b^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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