Optimal. Leaf size=191 \[ \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac {a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}+\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
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Rubi [A]
time = 0.36, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3713, 3686,
3716, 3709, 3610, 3612, 3556} \begin {gather*} \frac {a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (2 a C+3 b B) \cot ^3(c+d x)}{6 d}+\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \log (\sin (c+d x))}{d}+x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3686
Rule 3709
Rule 3713
Rule 3716
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^5(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \left (2 a (3 b B+2 a C)-4 \left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)-2 b (a B-2 b C) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot ^3(c+d x) \left (-2 a \left (2 a^2 B-5 b^2 B-6 a b C\right )-4 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-2 b^2 (a B-2 b C) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot ^2(c+d x) \left (-4 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right )+4 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac {a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac {1}{4} \int \cot (c+d x) \left (4 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right )+4 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)\right ) \, dx\\ &=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac {a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \int \cot (c+d x) \, dx\\ &=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac {a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac {a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}+\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.51, size = 199, normalized size = 1.04 \begin {gather*} \frac {12 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)+6 a \left (a^2 B-3 b^2 B-3 a b C\right ) \cot ^2(c+d x)-4 a^2 (3 b B+a C) \cot ^3(c+d x)-3 a^3 B \cot ^4(c+d x)-6 (a+i b)^3 (B+i C) \log (i-\tan (c+d x))+12 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \log (\tan (c+d x))-6 (a-i b)^3 (B-i C) \log (i+\tan (c+d x))}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 203, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {B \,b^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+C \,b^{3} \ln \left (\sin \left (d x +c \right )\right )+3 B a \,b^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 C a \,b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+3 B \,a^{2} b \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 C \,a^{2} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+B \,a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+C \,a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(203\) |
default | \(\frac {B \,b^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+C \,b^{3} \ln \left (\sin \left (d x +c \right )\right )+3 B a \,b^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+3 C a \,b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+3 B \,a^{2} b \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 C \,a^{2} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+B \,a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+C \,a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(203\) |
norman | \(\frac {\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {B \,a^{3} \tan \left (d x +c \right )}{4 d}+\frac {a \left (a^{2} B -3 b^{2} B -3 C a b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (3 B b +C a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}}{\tan \left (d x +c \right )^{5}}+\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(233\) |
risch | \(-\frac {2 i B \,a^{3} c}{d}+3 i B a \,b^{2} x -i B \,a^{3} x +3 i C \,a^{2} b x +3 B \,a^{2} b x -B \,b^{3} x +C \,a^{3} x -3 C a \,b^{2} x +\frac {6 i b C \,a^{2} c}{d}-i C \,b^{3} x +\frac {6 i B a \,b^{2} c}{d}-\frac {2 i \left (-9 C a \,b^{2}+4 C \,a^{3}+12 B \,a^{2} b -3 B \,b^{3}-6 i B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+6 i B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-30 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-18 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+36 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 i B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+9 i C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-18 i C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 i C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+3 B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+9 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-9 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-18 i B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+9 i B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+9 i B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-27 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+9 C a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+27 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-10 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {2 i C \,b^{3} c}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B a \,b^{2}}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C b}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C \,b^{3}}{d}\) | \(577\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 215, normalized size = 1.13 \begin {gather*} \frac {12 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} - 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {3 \, B a^{3} - 12 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 4 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\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 [A]
time = 4.02, size = 225, normalized size = 1.18 \begin {gather*} \frac {6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\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 )^{4} + 3 \, {\left (3 \, B a^{3} - 6 \, C a^{2} b - 6 \, B a b^{2} + 4 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 3 \, B a^{3} + 12 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} - 4 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs.
\(2 (187) = 374\).
time = 6.51, size = 398, normalized size = 2.08 \begin {gather*} \begin {cases} \text {NaN} & \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 )^{3} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\- \frac {B a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {B a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 B a^{2} b x + \frac {3 B a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {B a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 B a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 B a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - B b^{3} x - \frac {B b^{3}}{d \tan {\left (c + d x \right )}} + C a^{3} x + \frac {C a^{3}}{d \tan {\left (c + d x \right )}} - \frac {C a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {3 C a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 C a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 C a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - 3 C a b^{2} x - \frac {3 C a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {C b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \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. 528 vs.
\(2 (185) = 370\).
time = 1.60, size = 528, normalized size = 2.76 \begin {gather*} -\frac {3 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 288 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 192 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} + 192 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 360 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3}}{\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 = 8.94, size = 204, normalized size = 1.07 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^3-3\,C\,a^2\,b-3\,B\,a\,b^2+C\,b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {C\,a^3}{3}+B\,b\,a^2\right )+\frac {B\,a^3}{4}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {B\,a^3}{2}+\frac {3\,C\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-C\,a^3-3\,B\,a^2\,b+3\,C\,a\,b^2+B\,b^3\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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