Optimal. Leaf size=45 \[ -\frac {4 e^{(2 i+m) x} \, _2F_1\left (2,1-\frac {i m}{2};2-\frac {i m}{2};e^{2 i x}\right )}{2 i+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4538}
\begin {gather*} -\frac {4 e^{(m+2 i) x} \text {Hypergeometric2F1}\left (2,1-\frac {i m}{2},2-\frac {i m}{2},e^{2 i x}\right )}{m+2 i} \end {gather*}
Antiderivative was successfully verified.
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Rule 4538
Rubi steps
\begin {align*} \int e^{m x} \csc ^2(x) \, dx &=-\frac {4 e^{(2 i+m) x} \, _2F_1\left (2,1-\frac {i m}{2};2-\frac {i m}{2};e^{2 i x}\right )}{2 i+m}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 90, normalized size = 2.00 \begin {gather*} \frac {e^{m x} \left (e^{2 i x} m \, _2F_1\left (1,1-\frac {i m}{2};2-\frac {i m}{2};e^{2 i x}\right )+(2 i+m) \left (-i \cot (x)+\, _2F_1\left (1,-\frac {i m}{2};1-\frac {i m}{2};e^{2 i x}\right )\right )\right )}{-2+i m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{m x}}{\sin \left (x \right )^{2}}\, dx\]
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^{m x}}{\sin ^{2}{\left (x \right )}}\, 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}}^{m\,x}}{{\sin \left (x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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