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 = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 37
Rule 3088
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*}
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Mathematica [B] time = 0.06, 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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fricas [A] time = 0.67, size = 30, normalized size = 0.94 \[ \frac {{\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 57, normalized size = 1.78 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 23, normalized size = 0.72 \[ \frac {i}{2 d \,a^{3} \left (i \tan \left (d x +c \right )+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 51, normalized size = 1.59 \[ \frac {i \, \cos \left (4 \, d x + 4 \, c\right ) + 2 i \, \cos \left (2 \, d x + 2 \, c\right ) + \sin \left (4 \, d x + 4 \, c\right ) + 2 \, \sin \left (2 \, d x + 2 \, c\right )}{8 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 100, normalized size = 3.12 \[ -\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,1{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )}{a^3\,d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,1{}\mathrm {i}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,6{}\mathrm {i}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 97, normalized size = 3.03 \[ \begin {cases} \frac {\left (8 i a^{3} d e^{4 i c} e^{- 2 i d x} + 4 i a^{3} d e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{32 a^{6} d^{2}} & \text {for}\: 32 a^{6} d^{2} e^{6 i c} \neq 0 \\\frac {x \left (e^{2 i c} + 1\right ) e^{- 4 i c}}{2 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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