Optimal. Leaf size=163 \[ -\frac {a^4 (7 B+8 i A) \log (\sin (c+d x))}{d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.45, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3593, 3589, 3475, 3531} \[ -\frac {a^4 (7 B+8 i A) \log (\sin (c+d x))}{d}-\frac {(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3589
Rule 3593
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) (a+i a \tan (c+d x))^3 (3 a (2 i A+B)+3 i a B \tan (c+d x)) \, dx\\ &=-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {(2 i A+B) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {1}{6} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \left (-6 a^2 (4 A-3 i B)-6 a^2 B \tan (c+d x)\right ) \, dx\\ &=-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {(2 i A+B) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {1}{6} \int \cot (c+d x) (a+i a \tan (c+d x)) \left (-6 a^3 (8 i A+7 B)-6 i a^3 B \tan (c+d x)\right ) \, dx\\ &=-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {(2 i A+B) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {1}{6} \int \cot (c+d x) \left (-6 a^4 (8 i A+7 B)+48 a^4 (A-i B) \tan (c+d x)\right ) \, dx+\left (a^4 B\right ) \int \tan (c+d x) \, dx\\ &=8 a^4 (A-i B) x-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {(2 i A+B) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\left (a^4 (8 i A+7 B)\right ) \int \cot (c+d x) \, dx\\ &=8 a^4 (A-i B) x-\frac {a^4 B \log (\cos (c+d x))}{d}-\frac {a^4 (8 i A+7 B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac {(2 i A+B) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}\\ \end {align*}
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Mathematica [B] time = 10.61, size = 1138, normalized size = 6.98 \[ a^4 \left (\frac {x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left (40 A \cos ^4(c)-\frac {71}{2} i B \cos ^4(c)+8 i A \cot (c) \cos ^4(c)+7 B \cot (c) \cos ^4(c)-80 i A \sin (c) \cos ^3(c)-\frac {145}{2} B \sin (c) \cos ^3(c)-80 A \sin ^2(c) \cos ^2(c)+75 i B \sin ^2(c) \cos ^2(c)+\frac {1}{2} i B \cos ^2(c)+40 i A \sin ^3(c) \cos (c)+40 B \sin ^3(c) \cos (c)+\frac {3}{2} B \sin (c) \cos (c)+8 A \sin ^4(c)-\frac {19}{2} i B \sin ^4(c)-\frac {3}{2} i B \sin ^2(c)-i (4 \cos (2 c) A+4 A-3 i B-4 i B \cos (2 c)) \csc (c) \sec (c) (\cos (4 c)-i \sin (4 c))-\frac {1}{2} B \sin ^4(c) \tan (c)-\frac {1}{2} B \sin ^2(c) \tan (c)\right ) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}-\frac {B \cos (4 c) (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \log \left (\cos ^2(c+d x)\right ) \sin ^5(c+d x)}{2 d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 A \cos (2 c)-7 i B \cos (2 c)-8 i A \sin (2 c)-7 B \sin (2 c)) \left (i \tan ^{-1}(\tan (5 c+d x)) \sin (2 c)-\tan ^{-1}(\tan (5 c+d x)) \cos (2 c)\right ) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 A \cos (2 c)-7 i B \cos (2 c)-8 i A \sin (2 c)-7 B \sin (2 c)) \left (-\frac {1}{2} i \cos (2 c) \log \left (\sin ^2(c+d x)\right )-\frac {1}{2} \sin (2 c) \log \left (\sin ^2(c+d x)\right )\right ) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {i B (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \log \left (\cos ^2(c+d x)\right ) \sin (4 c) \sin ^5(c+d x)}{2 d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(A-i B) (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (8 d x \cos (4 c)-8 i d x \sin (4 c)) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \left (\frac {2}{3} i \sin (4 c)-\frac {2}{3} \cos (4 c)\right ) (11 A \sin (d x)-6 i B \sin (d x)) \sin ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) (-2 A \cos (c)-12 i A \sin (c)-3 B \sin (c)) \left (\frac {1}{6} \cos (4 c)-\frac {1}{6} i \sin (4 c)\right ) \sin ^3(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {A (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \left (\frac {1}{3} \cos (4 c)-\frac {1}{3} i \sin (4 c)\right ) \sin (d x) \sin ^2(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 245, normalized size = 1.50 \[ \frac {{\left (72 i \, A + 30 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-108 i \, A - 54 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (44 i \, A + 24 \, B\right )} a^{4} - 3 \, {\left (B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - B a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + {\left ({\left (-24 i \, A - 21 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (72 i \, A + 63 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-72 i \, A - 63 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (24 i \, A + 21 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.49, size = 291, normalized size = 1.79 \[ \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, B a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 24 \, B a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - 87 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, {\left (8 i \, A a^{4} + 8 \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 24 \, {\left (8 i \, A a^{4} + 7 \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {-352 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 308 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 87 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 170, normalized size = 1.04 \[ 8 A \,a^{4} x +\frac {8 A \,a^{4} c}{d}-\frac {a^{4} B \ln \left (\cos \left (d x +c \right )\right )}{d}-\frac {8 i B \,a^{4} c}{d}-\frac {2 i A \,a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{d}-\frac {8 i A \,a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {7 A \cot \left (d x +c \right ) a^{4}}{d}-\frac {7 a^{4} B \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {4 i B \cot \left (d x +c \right ) a^{4}}{d}-8 i B x \,a^{4}-\frac {A \,a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{4} B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 117, normalized size = 0.72 \[ \frac {48 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} + 6 \, {\left (4 i \, A + 4 \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, {\left (-8 i \, A - 7 \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - {\left (12 i \, A + 3 \, B\right )} a^{4} \tan \left (d x + c\right ) - 2 \, A a^{4}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.71, size = 113, normalized size = 0.69 \[ -\frac {\frac {A\,a^4}{3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (7\,A\,a^4-B\,a^4\,4{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{2}+A\,a^4\,2{}\mathrm {i}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3}-\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (7\,B+A\,8{}\mathrm {i}\right )}{d}+\frac {8\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.30, size = 294, normalized size = 1.80 \[ - \frac {B a^{4} \log {\left (\frac {4 i A a^{4} + 3 B a^{4}}{4 i A a^{4} e^{2 i c} + 3 B a^{4} e^{2 i c}} + e^{2 i d x} \right )}}{d} - \frac {i a^{4} \left (8 A - 7 i B\right ) \log {\left (e^{2 i d x} + \frac {4 i A a^{4} + 4 B a^{4} - i a^{4} \left (8 A - 7 i B\right )}{4 i A a^{4} e^{2 i c} + 3 B a^{4} e^{2 i c}} \right )}}{d} + \frac {44 i A a^{4} + 24 B a^{4} + \left (- 108 i A a^{4} e^{2 i c} - 54 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (72 i A a^{4} e^{4 i c} + 30 B a^{4} e^{4 i c}\right ) e^{4 i d x}}{3 d e^{6 i c} e^{6 i d x} - 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} - 3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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