Optimal. Leaf size=15 \[ -\frac {e^x \sin (x)}{1-\cos (x)} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2288} \[ -\frac {e^x \sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2288
Rubi steps
\begin {align*} \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx &=-\frac {e^x \sin (x)}{1-\cos (x)}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 11, normalized size = 0.73 \[ -e^x \cot \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.15, size = 12, normalized size = 0.80 \[ -\frac {{\left (\cos \relax (x) + 1\right )} e^{x}}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 10, normalized size = 0.67 \[ -\frac {e^{x}}{\tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 21, normalized size = 1.40
method | result | size |
risch | \(-i {\mathrm e}^{x}-\frac {2 i {\mathrm e}^{x}}{{\mathrm e}^{i x}-1}\) | \(21\) |
norman | \(\frac {-{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 22, normalized size = 1.47 \[ -\frac {2 \, e^{x} \sin \relax (x)}{\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 8, normalized size = 0.53 \[ -\mathrm {cot}\left (\frac {x}{2}\right )\,{\mathrm {e}}^x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sin {\relax (x )} - 1\right ) e^{x}}{\cos {\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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