Optimal. Leaf size=42 \[ (2-2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right )-\frac {e^x \sin (x)}{1+\cos (x)} \]
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Rubi [A]
time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.05, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4551, 4548,
4527, 2225, 2283, 2326} \begin {gather*} -4 i e^x \text {Hypergeometric2F1}\left (-i,1,1-i,-e^{i x}\right )+2 i e^x+\frac {e^x \sin (x)}{\cos (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 2326
Rule 4527
Rule 4548
Rule 4551
Rubi steps
\begin {align*} \int \frac {e^x (1-\sin (x))}{1+\cos (x)} \, dx &=-\left (2 \int \frac {e^x \sin (x)}{1+\cos (x)} \, dx\right )+\int \frac {e^x (1+\sin (x))}{1+\cos (x)} \, dx\\ &=\frac {e^x \sin (x)}{1+\cos (x)}-2 \int e^x \tan \left (\frac {x}{2}\right ) \, dx\\ &=\frac {e^x \sin (x)}{1+\cos (x)}-2 i \int \left (-e^x+\frac {2 e^x}{1+e^{i x}}\right ) \, dx\\ &=\frac {e^x \sin (x)}{1+\cos (x)}+2 i \int e^x \, dx-4 i \int \frac {e^x}{1+e^{i x}} \, dx\\ &=2 i e^x-4 i e^x \, _2F_1\left (-i,1;1-i;-e^{i x}\right )+\frac {e^x \sin (x)}{1+\cos (x)}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(87\) vs. \(2(42)=84\).
time = 0.16, size = 87, normalized size = 2.07 \begin {gather*} -\frac {2 e^x \cos \left (\frac {x}{2}\right ) \left (2 i \cos \left (\frac {x}{2}\right ) \, _2F_1\left (-i,1;1-i;-e^{i x}\right )-(1+i) e^{i x} \cos \left (\frac {x}{2}\right ) \, _2F_1\left (1,1-i;2-i;-e^{i x}\right )-\sin \left (\frac {x}{2}\right )\right )}{1+\cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} \left (1-\sin \left (x \right )\right )}{1+\cos \left (x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \int \left (- \frac {e^{x}}{\cos {\left (x \right )} + 1}\right )\, dx - \int \frac {e^{x} \sin {\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {{\mathrm {e}}^x\,\left (\sin \left (x\right )-1\right )}{\cos \left (x\right )+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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