Optimal. Leaf size=46 \[ (2+2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-i e^{i x}\right )-\frac {e^x \cos (x)}{1-\sin (x)} \]
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Rubi [A]
time = 0.09, antiderivative size = 48, normalized size of antiderivative = 1.04, number of steps
used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4550, 4547,
4527, 2225, 2283, 2326} \begin {gather*} -4 i e^x \text {Hypergeometric2F1}\left (-i,1,1-i,-i e^{i x}\right )+2 i e^x+\frac {e^x \cos (x)}{1-\sin (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 2326
Rule 4527
Rule 4547
Rule 4550
Rubi steps
\begin {align*} \int \frac {e^x (1-\cos (x))}{1-\sin (x)} \, dx &=-\left (2 \int \frac {e^x \cos (x)}{1-\sin (x)} \, dx\right )+\int \frac {e^x (1+\cos (x))}{1-\sin (x)} \, dx\\ &=\frac {e^x \cos (x)}{1-\sin (x)}-2 \int e^x \tan \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx\\ &=\frac {e^x \cos (x)}{1-\sin (x)}-2 i \int \left (-e^x+\frac {2 e^x}{1+e^{2 i \left (\frac {\pi }{4}+\frac {x}{2}\right )}}\right ) \, dx\\ &=\frac {e^x \cos (x)}{1-\sin (x)}+2 i \int e^x \, dx-4 i \int \frac {e^x}{1+e^{2 i \left (\frac {\pi }{4}+\frac {x}{2}\right )}} \, dx\\ &=2 i e^x-4 i e^x \, _2F_1\left (-i,1;1-i;-i e^{i x}\right )+\frac {e^x \cos (x)}{1-\sin (x)}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 72, normalized size = 1.57 \begin {gather*} \frac {1}{2} (-1+\cos (x)) \csc ^2\left (\frac {x}{2}\right ) \left (-\frac {e^x \left ((1-2 i)+(1+2 i) \cot \left (\frac {x}{2}\right )\right )}{-1+\cot \left (\frac {x}{2}\right )}+4 i \, _2F_1(-i,1;1-i;-i \cos (x)+\sin (x)) (\cosh (x)+\sinh (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} \left (1-\cos \left (x \right )\right )}{1-\sin \left (x \right )}\, dx\]
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 \frac {\left (\cos {\left (x \right )} - 1\right ) 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 = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (\cos \left (x\right )-1\right )}{\sin \left (x\right )-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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