Optimal. Leaf size=93 \[ 2 i a^2 x+\frac {2 i a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^2(c+d x)}{d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.10, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3623, 3610,
3612, 3556} \begin {gather*} -\frac {a^2 \cot ^4(c+d x)}{4 d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^2(c+d x)}{d}+\frac {2 i a^2 \cot (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d}+2 i a^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3623
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {a^2 \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) \left (-2 a^2-2 i a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {a^2 \cot ^2(c+d x)}{d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) \left (-2 i a^2+2 a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {2 i a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^2(c+d x)}{d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) \left (2 a^2+2 i a^2 \tan (c+d x)\right ) \, dx\\ &=2 i a^2 x+\frac {2 i a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^2(c+d x)}{d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\left (2 a^2\right ) \int \cot (c+d x) \, dx\\ &=2 i a^2 x+\frac {2 i a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^2(c+d x)}{d}-\frac {2 i a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^4(c+d x)}{4 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.41, size = 79, normalized size = 0.85 \begin {gather*} -\frac {a^2 \left (8 i \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )+3 \left (-4 \cot ^2(c+d x)+\cot ^4(c+d x)-8 (\log (\cos (c+d x))+\log (\tan (c+d x)))\right )\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 90, normalized size = 0.97
method | result | size |
risch | \(-\frac {4 i a^{2} c}{d}-\frac {2 a^{2} \left (21 \,{\mathrm e}^{6 i \left (d x +c \right )}-36 \,{\mathrm e}^{4 i \left (d x +c \right )}+29 \,{\mathrm e}^{2 i \left (d x +c \right )}-8\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(88\) |
derivativedivides | \(\frac {-a^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 i a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(90\) |
default | \(\frac {-a^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 i a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(90\) |
norman | \(\frac {\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{4 d}-\frac {2 i a^{2} \tan \left (d x +c \right )}{3 d}+\frac {2 i a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{d}+2 i a^{2} x \left (\tan ^{4}\left (d x +c \right )\right )}{\tan \left (d x +c \right )^{4}}+\frac {2 a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 97, normalized size = 1.04 \begin {gather*} -\frac {-24 i \, {\left (d x + c\right )} a^{2} + 12 \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 \, a^{2} \log \left (\tan \left (d x + c\right )\right ) - \frac {24 i \, a^{2} \tan \left (d x + c\right )^{3} + 12 \, a^{2} \tan \left (d x + c\right )^{2} - 8 i \, a^{2} \tan \left (d x + c\right ) - 3 \, a^{2}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 174 vs. \(2 (83) = 166\).
time = 0.42, size = 174, normalized size = 1.87 \begin {gather*} -\frac {2 \, {\left (21 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 36 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 29 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \, a^{2} - 3 \, {\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.98, size = 168, normalized size = 1.81 \begin {gather*} \frac {2 a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 42 a^{2} e^{6 i c} e^{6 i d x} + 72 a^{2} e^{4 i c} e^{4 i d x} - 58 a^{2} e^{2 i c} e^{2 i d x} + 16 a^{2}}{3 d e^{8 i c} e^{8 i d x} - 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} - 12 d e^{2 i c} e^{2 i d x} + 3 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. 180 vs. \(2 (83) = 166\).
time = 0.95, size = 180, normalized size = 1.94 \begin {gather*} -\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 768 \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 384 \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 240 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {800 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.92, size = 80, normalized size = 0.86 \begin {gather*} \frac {a^2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{d}-\frac {-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\,2{}\mathrm {i}-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,2{}\mathrm {i}}{3}+\frac {a^2}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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