Optimal. Leaf size=29 \[ \frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3071} \[ \frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin {align*} \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=\frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.00 \[ \frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 17, normalized size = 0.59 \[ \frac {i \, e^{\left (-i \, d x - i \, c\right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 0.72 \[ \frac {2}{a d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 23, normalized size = 0.79 \[ \frac {2}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 29, normalized size = 1.00 \[ \frac {2}{{\left (-i \, a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 25, normalized size = 0.86 \[ \frac {2{}\mathrm {i}}{a\,d\,\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 31, normalized size = 1.07 \[ \begin {cases} \frac {i e^{- i c} e^{- i d x}}{a d} & \text {for}\: a d e^{i c} \neq 0 \\\frac {x e^{- i c}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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