Optimal. Leaf size=38 \[ -\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \]
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Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3542, 3531, 3475} \[ -\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3542
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {a^2 \cot (c+d x)}{d}+\int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=-2 a^2 x-\frac {a^2 \cot (c+d x)}{d}+\left (2 i a^2\right ) \int \cot (c+d x) \, dx\\ &=-2 a^2 x-\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [B] time = 0.79, size = 100, normalized size = 2.63 \[ \frac {a^2 \csc (c) \csc (c+d x) \left (4 d x \cos (2 c+d x)+4 \sin (c) \sin (c+d x) \tan ^{-1}(\tan (3 c+d x))-i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )+\cos (d x) \left (-4 d x+i \log \left (\sin ^2(c+d x)\right )\right )+2 \sin (d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 57, normalized size = 1.50 \[ \frac {-2 i \, a^{2} + {\left (2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 85, normalized size = 2.24 \[ -\frac {8 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 4 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-4 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 47, normalized size = 1.24 \[ \frac {2 i a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-2 a^{2} x -\frac {a^{2} \cot \left (d x +c \right )}{d}-\frac {2 a^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 56, normalized size = 1.47 \[ -\frac {2 \, {\left (d x + c\right )} a^{2} + i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac {a^{2}}{\tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.83, size = 29, normalized size = 0.76 \[ -\frac {a^2\,\left (\mathrm {cot}\left (c+d\,x\right )+4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 51, normalized size = 1.34 \[ \frac {2 i a^{2}}{- d e^{2 i c} e^{2 i d x} + d} + \frac {2 i a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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