Optimal. Leaf size=38 \[ -2 a^2 x-\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.05, 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 = {3623, 3612,
3556} \begin {gather*} -\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3612
Rule 3623
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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(100\) vs. \(2(38)=76\).
time = 0.56, size = 100, normalized size = 2.63 \begin {gather*} \frac {a^2 \csc (c) \csc (c+d x) \left (4 d x \cos (2 c+d x)+\cos (d x) \left (-4 d x+i \log \left (\sin ^2(c+d x)\right )\right )-i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )+2 \sin (d x)+4 \text {ArcTan}(\tan (3 c+d x)) \sin (c) \sin (c+d x)\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 49, normalized size = 1.29
method | result | size |
derivativedivides | \(\frac {-a^{2} \left (d x +c \right )+2 i a^{2} \ln \left (\sin \left (d x +c \right )\right )+a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(49\) |
default | \(\frac {-a^{2} \left (d x +c \right )+2 i a^{2} \ln \left (\sin \left (d x +c \right )\right )+a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(49\) |
risch | \(\frac {4 a^{2} c}{d}-\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(54\) |
norman | \(\frac {-\frac {a^{2}}{d}-2 a^{2} x \tan \left (d x +c \right )}{\tan \left (d x +c \right )}+\frac {2 i a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 56, normalized size = 1.47 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 58, normalized size = 1.53 \begin {gather*} -\frac {2 \, {\left (i \, a^{2} + {\left (-i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 51, normalized size = 1.34 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 85 vs. \(2 (36) = 72\).
time = 0.63, size = 85, normalized size = 2.24 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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