Optimal. Leaf size=198 \[ -4 a b \left (a^2-b^2\right ) x-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d} \]
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Rubi [A]
time = 0.30, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3646, 3716,
3709, 3610, 3612, 3556} \begin {gather*} -\frac {7 a^3 b \cot ^5(c+d x)}{15 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-4 a b x \left (a^2-b^2\right )-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3646
Rule 3709
Rule 3716
Rubi steps
\begin {align*} \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^6(c+d x) (a+b \tan (c+d x)) \left (14 a^2 b-6 a \left (a^2-3 b^2\right ) \tan (c+d x)-2 b \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^5(c+d x) \left (-2 a^2 \left (3 a^2-16 b^2\right )-24 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^4(c+d x) \left (-24 a b \left (a^2-b^2\right )+6 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^3(c+d x) \left (6 \left (a^4-6 a^2 b^2+b^4\right )+24 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^2(c+d x) \left (24 a b \left (a^2-b^2\right )-6 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot (c+d x) \left (-6 \left (a^4-6 a^2 b^2+b^4\right )-24 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\left (-a^4+6 a^2 b^2-b^4\right ) \int \cot (c+d x) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.51, size = 178, normalized size = 0.90 \begin {gather*} -\frac {4 a (a-b) b (a+b) \cot (c+d x)+\frac {1}{2} \left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)-\frac {4}{3} a (a-b) b (a+b) \cot ^3(c+d x)-\frac {1}{4} a^2 \left (a^2-6 b^2\right ) \cot ^4(c+d x)+\frac {4}{5} a^3 b \cot ^5(c+d x)+\frac {1}{6} a^4 \cot ^6(c+d x)-\frac {1}{2} (a-i b)^4 \log (i-\cot (c+d x))-\frac {1}{2} (a+i b)^4 \log (i+\cot (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 179, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{3} b \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+6 a^{2} b^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(179\) |
default | \(\frac {a^{4} \left (-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{3} b \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+6 a^{2} b^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 a \,b^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(179\) |
norman | \(\frac {-\frac {a^{4}}{6 d}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \left (a^{2}-6 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {4 a^{3} b \tan \left (d x +c \right )}{5 d}+\frac {4 a b \left (a^{2}-b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {4 a b \left (a^{2}-b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{d}-4 a b \left (a^{2}-b^{2}\right ) x \left (\tan ^{6}\left (d x +c \right )\right )}{\tan \left (d x +c \right )^{6}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(216\) |
risch | \(-4 a^{3} b x +4 a \,b^{3} x +i a^{4} x -6 i a^{2} b^{2} x +i b^{4} x +\frac {2 i a^{4} c}{d}-\frac {12 i a^{2} b^{2} c}{d}+\frac {2 i b^{4} c}{d}+\frac {72 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+112 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}-96 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+16 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-12 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-8 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-8 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-12 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+\frac {68 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+2 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-24 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-\frac {32 i a \,b^{3}}{3}+\frac {184 i a^{3} b}{15}-24 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-96 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+72 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+72 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {320 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{3}-\frac {368 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}-24 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}-64 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-\frac {248 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{5}+48 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{4}}{d}\) | \(554\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 195, normalized size = 0.98 \begin {gather*} -\frac {240 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {240 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{5} + 48 \, a^{3} b \tan \left (d x + c\right ) + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, a^{4} - 80 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\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 = 1.41, size = 214, normalized size = 1.08 \begin {gather*} -\frac {30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{6} + 5 \, {\left (11 \, a^{4} - 54 \, a^{2} b^{2} + 6 \, b^{4} + 48 \, {\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{6} + 240 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{5} + 48 \, a^{3} b \tan \left (d x + c\right ) + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, a^{4} - 80 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{60 \, d \tan \left (d x + c\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.67, size = 340, normalized size = 1.72 \begin {gather*} \begin {cases} \tilde {\infty } a^{4} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{7}{\left (c \right )} & \text {for}\: d = 0 \\\frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac {a^{4}}{4 d \tan ^{4}{\left (c + d x \right )}} - \frac {a^{4}}{6 d \tan ^{6}{\left (c + d x \right )}} - 4 a^{3} b x - \frac {4 a^{3} b}{d \tan {\left (c + d x \right )}} + \frac {4 a^{3} b}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {4 a^{3} b}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 a^{2} b^{2}}{d \tan ^{2}{\left (c + d x \right )}} - \frac {3 a^{2} b^{2}}{2 d \tan ^{4}{\left (c + d x \right )}} + 4 a b^{3} x + \frac {4 a b^{3}}{d \tan {\left (c + d x \right )}} - \frac {4 a b^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {b^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {b^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 506 vs.
\(2 (188) = 376\).
time = 1.37, size = 506, normalized size = 2.56 \begin {gather*} -\frac {5 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 435 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2160 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5280 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7680 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - 1920 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 1920 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4704 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 28224 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4704 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 5280 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 435 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a^{3} b \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 = 4.25, size = 202, normalized size = 1.02 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^6\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {4\,a\,b^3}{3}-\frac {4\,a^3\,b}{3}\right )-{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (4\,a\,b^3-4\,a^3\,b\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{4}-\frac {3\,a^2\,b^2}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {a^4}{2}-3\,a^2\,b^2+\frac {b^4}{2}\right )+\frac {a^4}{6}+\frac {4\,a^3\,b\,\mathrm {tan}\left (c+d\,x\right )}{5}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^4-6\,a^2\,b^2+b^4\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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