Optimal. Leaf size=163 \[ -\frac{a^4 (7 B+8 i A) \log (\sin (c+d x))}{d}-\frac{(B+2 i A) \cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac{(4 A-3 i B) \cot (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+8 a^4 x (A-i B)-\frac{a^4 B \log (\cos (c+d x))}{d}-\frac{a A \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.452709, antiderivative size = 163, normalized size of antiderivative = 1., 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 3593
Rule 3589
Rule 3475
Rule 3531
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*}
Mathematica [B] time = 9.59529, 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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Maple [A] time = 0.074, size = 170, normalized size = 1. \begin{align*} 8\,A{a}^{4}x+8\,{\frac{A{a}^{4}c}{d}}-{\frac{B{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-8\,iBx{a}^{4}-{\frac{8\,iB{a}^{4}c}{d}}-{\frac{2\,iA{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+7\,{\frac{A\cot \left ( dx+c \right ){a}^{4}}{d}}-7\,{\frac{B{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{8\,iA{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,iB\cot \left ( dx+c \right ){a}^{4}}{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.37657, size = 159, normalized size = 0.98 \begin{align*} \frac{6 \,{\left (d x + c\right )}{\left (8 \, A - 8 i \, B\right )} a^{4} - 24 \,{\left (-i \, A - 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{{\left (42 \, A - 24 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 3 \,{\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) - 2 \, A a^{4}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53453, size = 679, normalized size = 4.17 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 28.4718, size = 262, normalized size = 1.61 \begin{align*} \frac{\frac{\left (24 i A a^{4} + 10 B a^{4}\right ) e^{- 2 i c} e^{4 i d x}}{d} - \frac{\left (36 i A a^{4} + 18 B a^{4}\right ) e^{- 4 i c} e^{2 i d x}}{d} + \frac{\left (44 i A a^{4} + 24 B a^{4}\right ) e^{- 6 i c}}{3 d}}{e^{6 i d x} - 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} - e^{- 6 i c}} + \operatorname{RootSum}{\left (z^{2} d^{2} + z \left (8 i A a^{4} d + 8 B a^{4} d\right ) + 8 i A B a^{8} + 7 B^{2} a^{8}, \left ( i \mapsto i \log{\left (- \frac{i i d}{4 A a^{4} e^{2 i c} - 3 i B a^{4} e^{2 i c}} + \frac{4 A - 4 i B}{4 A e^{2 i c} - 3 i B e^{2 i c}} + e^{2 i d x} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.90507, size = 397, normalized size = 2.44 \begin{align*} \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 ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, B a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\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 ) - 384 \,{\left (-i \, A a^{4} - 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 ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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