Optimal. Leaf size=100 \[ -a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {i a \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3610, 3612,
3556} \begin {gather*} -\frac {a \cot ^5(c+d x)}{5 d}-\frac {i a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^3(c+d x)}{3 d}+\frac {i a \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x)}{d}+\frac {i a \log (\sin (c+d x))}{d}-a x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) (i a-a \tan (c+d x)) \, dx\\ &=-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) (-a-i a \tan (c+d x)) \, dx\\ &=\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) (-i a+a \tan (c+d x)) \, dx\\ &=\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) (i a-a \tan (c+d x)) \, dx\\ &=-a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+(i a) \int \cot (c+d x) \, dx\\ &=-a x-\frac {a \cot (c+d x)}{d}+\frac {i a \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {i a \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {i a \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 = 0.38, size = 84, normalized size = 0.84 \begin {gather*} -\frac {a \cot ^5(c+d x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac {i a \left (2 \cot ^2(c+d x)-\cot ^4(c+d x)+4 \log (\cos (c+d x))+4 \log (\tan (c+d x))\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 76, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {i a \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(76\) |
default | \(\frac {i a \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(76\) |
risch | \(\frac {2 a c}{d}-\frac {2 i a \left (75 \,{\mathrm e}^{8 i \left (d x +c \right )}-150 \,{\mathrm e}^{6 i \left (d x +c \right )}+200 \,{\mathrm e}^{4 i \left (d x +c \right )}-100 \,{\mathrm e}^{2 i \left (d x +c \right )}+23\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(94\) |
norman | \(\frac {-\frac {a}{5 d}-a x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {i a \tan \left (d x +c \right )}{4 d}+\frac {i a \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {i a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 93, normalized size = 0.93 \begin {gather*} -\frac {60 \, {\left (d x + c\right )} a + 30 i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a \tan \left (d x + c\right )^{4} - 30 i \, a \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 i \, a \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, 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. 197 vs. \(2 (86) = 172\).
time = 0.51, size = 197, normalized size = 1.97 \begin {gather*} \frac {-150 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 300 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 400 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 200 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 15 \, {\left (-i \, a e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 46 i \, a}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, 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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 206 vs. \(2 (83) = 166\).
time = 0.30, size = 206, normalized size = 2.06 \begin {gather*} \frac {i a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 150 i a e^{8 i c} e^{8 i d x} + 300 i a e^{6 i c} e^{6 i d x} - 400 i a e^{4 i c} e^{4 i d x} + 200 i a e^{2 i c} e^{2 i d x} - 46 i a}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 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. 186 vs. \(2 (86) = 172\).
time = 0.67, size = 186, normalized size = 1.86 \begin {gather*} \frac {6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1920 i \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 960 i \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {-2192 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.07, size = 79, normalized size = 0.79 \begin {gather*} -\frac {2\,a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {1{}\mathrm {i}\,a\,\mathrm {tan}\left (c+d\,x\right )}{4}+\frac {a}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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