Optimal. Leaf size=162 \[ -8 i a^4 x-\frac {8 i a^4 \cot (c+d x)}{d}-\frac {4 a^4 \cot ^2(c+d x)}{d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {8 a^4 \log (\sin (c+d x))}{d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d} \]
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Rubi [A]
time = 0.22, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3634, 3674,
3672, 3610, 3612, 3556} \begin {gather*} \frac {67 a^4 \cot ^4(c+d x)}{60 d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}-\frac {4 a^4 \cot ^2(c+d x)}{d}-\frac {8 i a^4 \cot (c+d x)}{d}-\frac {8 a^4 \log (\sin (c+d x))}{d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-8 i a^4 x-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3634
Rule 3672
Rule 3674
Rubi steps
\begin {align*} \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {1}{6} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \left (-14 i a^2+10 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac {1}{30} \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \left (134 a^3+106 i a^3 \tan (c+d x)\right ) \, dx\\ &=\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac {1}{30} \int \cot ^4(c+d x) \left (240 i a^4-240 a^4 \tan (c+d x)\right ) \, dx\\ &=\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac {1}{30} \int \cot ^3(c+d x) \left (-240 a^4-240 i a^4 \tan (c+d x)\right ) \, dx\\ &=-\frac {4 a^4 \cot ^2(c+d x)}{d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac {1}{30} \int \cot ^2(c+d x) \left (-240 i a^4+240 a^4 \tan (c+d x)\right ) \, dx\\ &=-\frac {8 i a^4 \cot (c+d x)}{d}-\frac {4 a^4 \cot ^2(c+d x)}{d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\frac {1}{30} \int \cot (c+d x) \left (240 a^4+240 i a^4 \tan (c+d x)\right ) \, dx\\ &=-8 i a^4 x-\frac {8 i a^4 \cot (c+d x)}{d}-\frac {4 a^4 \cot ^2(c+d x)}{d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 d}-\left (8 a^4\right ) \int \cot (c+d x) \, dx\\ &=-8 i a^4 x-\frac {8 i a^4 \cot (c+d x)}{d}-\frac {4 a^4 \cot ^2(c+d x)}{d}+\frac {8 i a^4 \cot ^3(c+d x)}{3 d}+\frac {67 a^4 \cot ^4(c+d x)}{60 d}-\frac {8 a^4 \log (\sin (c+d x))}{d}-\frac {\cot ^6(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{6 d}-\frac {7 i \cot ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{15 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. \(363\) vs. \(2(162)=324\).
time = 1.39, size = 363, normalized size = 2.24 \begin {gather*} \frac {a^4 \csc (c) \csc ^6(c+d x) \left (860 i \cos (c)-780 i \cos (c+2 d x)-510 i \cos (3 c+2 d x)+366 i \cos (3 c+4 d x)+150 i \cos (5 c+4 d x)-86 i \cos (5 c+6 d x)-490 \sin (c)-600 i d x \sin (c)-300 \log \left (\sin ^2(c+d x)\right ) \sin (c)-345 \sin (c+2 d x)-450 i d x \sin (c+2 d x)-225 \log \left (\sin ^2(c+d x)\right ) \sin (c+2 d x)+345 \sin (3 c+2 d x)+450 i d x \sin (3 c+2 d x)+225 \log \left (\sin ^2(c+d x)\right ) \sin (3 c+2 d x)+120 \sin (3 c+4 d x)+180 i d x \sin (3 c+4 d x)+90 \log \left (\sin ^2(c+d x)\right ) \sin (3 c+4 d x)-120 \sin (5 c+4 d x)-180 i d x \sin (5 c+4 d x)-90 \log \left (\sin ^2(c+d x)\right ) \sin (5 c+4 d x)-30 i d x \sin (5 c+6 d x)-15 \log \left (\sin ^2(c+d x)\right ) \sin (5 c+6 d x)+30 i d x \sin (7 c+6 d x)+15 \log \left (\sin ^2(c+d x)\right ) \sin (7 c+6 d x)\right )}{240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 176, normalized size = 1.09
method | result | size |
risch | \(\frac {16 i a^{4} c}{d}+\frac {4 a^{4} \left (270 \,{\mathrm e}^{10 i \left (d x +c \right )}-855 \,{\mathrm e}^{8 i \left (d x +c \right )}+1350 \,{\mathrm e}^{6 i \left (d x +c \right )}-1125 \,{\mathrm e}^{4 i \left (d x +c \right )}+486 \,{\mathrm e}^{2 i \left (d x +c \right )}-86\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(110\) |
norman | \(\frac {-\frac {a^{4}}{6 d}+\frac {7 a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {4 a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {4 i a^{4} \tan \left (d x +c \right )}{5 d}+\frac {8 i a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 i a^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{d}-8 i a^{4} x \left (\tan ^{6}\left (d x +c \right )\right )}{\tan \left (d x +c \right )^{6}}-\frac {8 a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {4 a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(150\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )-4 i a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )-6 a^{4} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 i a^{4} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{4} \left (-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(176\) |
default | \(\frac {a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )-4 i a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )-6 a^{4} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 i a^{4} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{4} \left (-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 123, normalized size = 0.76 \begin {gather*} -\frac {480 i \, {\left (d x + c\right )} a^{4} - 240 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac {-480 i \, a^{4} \tan \left (d x + c\right )^{5} - 240 \, a^{4} \tan \left (d x + c\right )^{4} + 160 i \, a^{4} \tan \left (d x + c\right )^{3} + 105 \, a^{4} \tan \left (d x + c\right )^{2} - 48 i \, a^{4} \tan \left (d x + c\right ) - 10 \, a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 254, normalized size = 1.57 \begin {gather*} \frac {4 \, {\left (270 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 855 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 1350 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 1125 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 486 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 86 \, a^{4} - 30 \, {\left (a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, 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 = 6.05, size = 246, normalized size = 1.52 \begin {gather*} - \frac {8 a^{4} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {1080 a^{4} e^{10 i c} e^{10 i d x} - 3420 a^{4} e^{8 i c} e^{8 i d x} + 5400 a^{4} e^{6 i c} e^{6 i d x} - 4500 a^{4} e^{4 i c} e^{4 i d x} + 1944 a^{4} e^{2 i c} e^{2 i d x} - 344 a^{4}}{15 d e^{12 i c} e^{12 i d x} - 90 d e^{10 i c} e^{10 i d x} + 225 d e^{8 i c} e^{8 i d x} - 300 d e^{6 i c} e^{6 i d x} + 225 d e^{4 i c} e^{4 i d x} - 90 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 [A]
time = 0.91, size = 245, normalized size = 1.51 \begin {gather*} -\frac {5 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2835 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30720 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 15360 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 10080 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {37632 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2835 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.68, size = 107, normalized size = 0.66 \begin {gather*} -\frac {a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,16{}\mathrm {i}}{d}-\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5\,8{}\mathrm {i}+4\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3\,8{}\mathrm {i}}{3}-\frac {7\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2}{4}+\frac {a^4\,\mathrm {tan}\left (c+d\,x\right )\,4{}\mathrm {i}}{5}+\frac {a^4}{6}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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