Optimal. Leaf size=32 \[ \frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2} \]
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Rubi [A] time = 0.0318248, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {3088, 37} \[ \frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 37
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x}{(i a+a x)^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{i \cot ^2(c+d x)}{2 a^3 d (i+\cot (c+d x))^2}\\ \end{align*}
Mathematica [B] time = 0.0593853, size = 77, normalized size = 2.41 \[ \frac{\sin (2 (c+d x))}{4 a^3 d}+\frac{\sin (4 (c+d x))}{8 a^3 d}+\frac{i \cos (2 (c+d x))}{4 a^3 d}+\frac{i \cos (4 (c+d x))}{8 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 23, normalized size = 0.7 \begin{align*}{\frac{{\frac{i}{2}}}{d{a}^{3} \left ( i\tan \left ( dx+c \right ) +1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15423, size = 69, normalized size = 2.16 \begin{align*} \frac{i \, \cos \left (4 \, d x + 4 \, c\right ) + 2 i \, \cos \left (2 \, d x + 2 \, c\right ) + \sin \left (4 \, d x + 4 \, c\right ) + 2 \, \sin \left (2 \, d x + 2 \, c\right )}{8 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.462933, size = 86, normalized size = 2.69 \begin{align*} \frac{{\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.419022, size = 97, normalized size = 3.03 \begin{align*} \begin{cases} \frac{\left (8 i a^{3} d e^{4 i c} e^{- 2 i d x} + 4 i a^{3} d e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{32 a^{6} d^{2}} & \text{for}\: 32 a^{6} d^{2} e^{6 i c} \neq 0 \\\frac{x \left (e^{2 i c} + 1\right ) e^{- 4 i c}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12068, size = 77, normalized size = 2.41 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{3} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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