Optimal. Leaf size=45 \[ -\frac {4 e^{(m+2 i) x} \, _2F_1\left (2,1-\frac {i m}{2};2-\frac {i m}{2};e^{2 i x}\right )}{m+2 i} \]
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Rubi [A] time = 0.02, 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 = {4453} \[ -\frac {4 e^{(m+2 i) x} \text {Hypergeometric2F1}\left (2,1-\frac {i m}{2},2-\frac {i m}{2},e^{2 i x}\right )}{m+2 i} \]
Antiderivative was successfully verified.
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Rule 4453
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 = 0.20, size = 90, normalized size = 2.00 \[ \frac {e^{m x} \left (m e^{2 i x} \, _2F_1\left (1,1-\frac {i m}{2};2-\frac {i m}{2};e^{2 i x}\right )+(m+2 i) \left (\, _2F_1\left (1,-\frac {i m}{2};1-\frac {i m}{2};e^{2 i x}\right )-i \cot (x)\right )\right )}{-2+i m} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \csc ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {e^{\left (m x\right )}}{\cos \relax (x)^{2} - 1}, x\right ) \]
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 \[ \int \frac {e^{\left (m x\right )}}{\sin \relax (x)^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{m x}}{\sin \relax (x )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\mathrm {e}}^{m\,x}}{{\sin \relax (x)}^2} \,d x \]
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 \[ \int \frac {e^{m x}}{\sin ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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