Optimal. Leaf size=30 \[ (-1+i) e^{(1-i) x} \, _2F_1\left (1+i,2;2+i;-i e^{-i x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4541, 4535}
\begin {gather*} (-1+i) e^{(1-i) x} \text {Hypergeometric2F1}\left (1+i,2,2+i,-i e^{-i x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 4535
Rule 4541
Rubi steps
\begin {align*} \int \frac {e^x}{1+\sin (x)} \, dx &=\frac {1}{2} \int e^x \sec ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx\\ &=(-1+i) e^{(1-i) x} \, _2F_1\left (1+i,2;2+i;-i e^{-i x}\right )\\ \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. \(61\) vs. \(2(30)=60\).
time = 0.29, size = 61, normalized size = 2.03 \begin {gather*} \frac {2 e^x \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}-(1-i) (1-(1-i) \, _2F_1(-i,1;1-i;i \cos (x)-\sin (x))) (\cosh (x)+\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x}}{\sin \left (x \right )+1}\, dx\]
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 \frac {e^{x}}{\sin {\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 = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^x}{\sin \left (x\right )+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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