Optimal. Leaf size=15 \[ -2 e^{\sin (x)}+2 e^{\sin (x)} \sin (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 2207, 2225}
\begin {gather*} 2 e^{\sin (x)} \sin (x)-2 e^{\sin (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rubi steps
\begin {align*} \int e^{\sin (x)} \sin (2 x) \, dx &=\text {Subst}\left (\int 2 e^x x \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int e^x x \, dx,x,\sin (x)\right )\\ &=2 e^{\sin (x)} \sin (x)-2 \text {Subst}\left (\int e^x \, dx,x,\sin (x)\right )\\ &=-2 e^{\sin (x)}+2 e^{\sin (x)} \sin (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 0.73 \begin {gather*} e^{\sin (x)} (-2+2 \sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 14, normalized size = 0.93
method | result | size |
derivativedivides | \(-2 \,{\mathrm e}^{\sin \left (x \right )}+2 \,{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\) | \(14\) |
default | \(-2 \,{\mathrm e}^{\sin \left (x \right )}+2 \,{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\) | \(14\) |
risch | \(-2 \,{\mathrm e}^{\sin \left (x \right )}+2 \,{\mathrm e}^{\sin \left (x \right )} \sin \left (x \right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.37, size = 9, normalized size = 0.60 \begin {gather*} 2 \, {\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 9, normalized size = 0.60 \begin {gather*} 2 \, {\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.97, size = 15, normalized size = 1.00 \begin {gather*} 2 e^{\sin {\left (x \right )}} \sin {\left (x \right )} - 2 e^{\sin {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 9, normalized size = 0.60 \begin {gather*} 2 \, {\left (\sin \left (x\right ) - 1\right )} e^{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 9, normalized size = 0.60 \begin {gather*} 2\,{\mathrm {e}}^{\sin \left (x\right )}\,\left (\sin \left (x\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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