Optimal. Leaf size=183 \[ -\frac{(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac{(-B+3 i A) \log (\sin (c+d x))}{a^3 d}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{x (25 A+7 i B)}{8 a^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.529629, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3596, 3529, 3531, 3475} \[ -\frac{(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac{(-B+3 i A) \log (\sin (c+d x))}{a^3 d}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{x (25 A+7 i B)}{8 a^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3596
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^2(c+d x) (a (7 A+i B)-4 a (i A-B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\cot ^2(c+d x) \left (3 a^2 (13 A+3 i B)-3 a^2 (11 i A-5 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot ^2(c+d x) \left (6 a^3 (25 A+7 i B)-48 a^3 (3 i A-B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{(25 A+7 i B) \cot (c+d x)}{8 a^3 d}+\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (-48 a^3 (3 i A-B)-6 a^3 (25 A+7 i B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{(25 A+7 i B) x}{8 a^3}-\frac{(25 A+7 i B) \cot (c+d x)}{8 a^3 d}+\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{(3 i A-B) \int \cot (c+d x) \, dx}{a^3}\\ &=-\frac{(25 A+7 i B) x}{8 a^3}-\frac{(25 A+7 i B) \cot (c+d x)}{8 a^3 d}-\frac{(3 i A-B) \log (\sin (c+d x))}{a^3 d}+\frac{(A+i B) \cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(11 A+5 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(3 A+i B) \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 6.98094, size = 1282, normalized size = 7.01 \[ \frac{\csc \left (\frac{c}{2}\right ) \csc (c+d x) \sec \left (\frac{c}{2}\right ) \sec ^2(c+d x) \left (\frac{1}{2} i A \cos (3 c-d x)-\frac{1}{2} i A \cos (3 c+d x)-\frac{1}{2} A \sin (3 c-d x)+\frac{1}{2} A \sin (3 c+d x)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{2 d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(5 B-7 i A) \cos (4 d x) \sec ^2(c+d x) \left (\frac{\cos (c)}{32}-\frac{1}{32} i \sin (c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(11 B-23 i A) \cos (2 d x) \sec ^2(c+d x) \left (\frac{\cos (c)}{16}+\frac{1}{16} i \sin (c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left (-3 i A \cos \left (\frac{3 c}{2}\right )+B \cos \left (\frac{3 c}{2}\right )+3 A \sin \left (\frac{3 c}{2}\right )+i B \sin \left (\frac{3 c}{2}\right )\right ) \left (\tan ^{-1}(\tan (d x)) \sin \left (\frac{3 c}{2}\right )-i \tan ^{-1}(\tan (d x)) \cos \left (\frac{3 c}{2}\right )\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{\sec ^2(c+d x) \left (-3 i A \cos \left (\frac{3 c}{2}\right )+B \cos \left (\frac{3 c}{2}\right )+3 A \sin \left (\frac{3 c}{2}\right )+i B \sin \left (\frac{3 c}{2}\right )\right ) \left (\frac{1}{2} \cos \left (\frac{3 c}{2}\right ) \log \left (\sin ^2(c+d x)\right )+\frac{1}{2} i \sin \left (\frac{3 c}{2}\right ) \log \left (\sin ^2(c+d x)\right )\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{x \sec ^2(c+d x) (-6 A \cos (c)-2 i B \cos (c)+3 i A \cot (c) \cos (c)-B \cot (c) \cos (c)-3 i A \sin (c)+B \sin (c)+(B-3 i A) \cot (c) (\cos (3 c)+i \sin (3 c))) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{(A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(B-i A) \cos (6 d x) \sec ^2(c+d x) \left (\frac{1}{48} \cos (3 c)-\frac{1}{48} i \sin (3 c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(25 A+7 i B) \sec ^2(c+d x) \left (-\frac{1}{8} d x \cos (3 c)-\frac{1}{8} i d x \sin (3 c)\right ) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(23 A+11 i B) \sec ^2(c+d x) \left (-\frac{\cos (c)}{16}-\frac{1}{16} i \sin (c)\right ) \sin (2 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(7 A+5 i B) \sec ^2(c+d x) \left (\frac{1}{32} i \sin (c)-\frac{\cos (c)}{32}\right ) \sin (4 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3}+\frac{(A+i B) \sec ^2(c+d x) \left (\frac{1}{48} i \sin (3 c)-\frac{1}{48} \cos (3 c)\right ) \sin (6 d x) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{d (A \cos (c+d x)+B \sin (c+d x)) (i \tan (c+d x) a+a)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.12, size = 252, normalized size = 1.4 \begin{align*} -{\frac{17\,A}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{7\,i}{8}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{49\,i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{{a}^{3}d}}-{\frac{15\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{16\,{a}^{3}d}}+{\frac{A}{6\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{{\frac{i}{6}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{{\frac{5\,i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{3\,B}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,{a}^{3}d}}-{\frac{{\frac{i}{16}}A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}}-{\frac{A}{{a}^{3}d\tan \left ( dx+c \right ) }}-{\frac{3\,iA\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{3}d}}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53771, size = 504, normalized size = 2.75 \begin{align*} -\frac{12 \,{\left (49 \, A + 15 i \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} -{\left (12 \,{\left (49 \, A + 15 i \, B\right )} d x - 330 i \, A + 66 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} -{\left (117 i \, A - 51 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (19 i \, A - 13 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left ({\left (-288 i \, A + 96 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (288 i \, A - 96 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 i \, A + 2 \, B}{96 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.1081, size = 294, normalized size = 1.61 \begin{align*} - \frac{2 i A e^{- 2 i c}}{a^{3} d \left (e^{2 i d x} - e^{- 2 i c}\right )} - \frac{\left (\begin{cases} 49 A x e^{6 i c} + \frac{23 i A e^{4 i c} e^{- 2 i d x}}{2 d} + \frac{7 i A e^{2 i c} e^{- 4 i d x}}{4 d} + \frac{i A e^{- 6 i d x}}{6 d} + 15 i B x e^{6 i c} - \frac{11 B e^{4 i c} e^{- 2 i d x}}{2 d} - \frac{5 B e^{2 i c} e^{- 4 i d x}}{4 d} - \frac{B e^{- 6 i d x}}{6 d} & \text{for}\: d \neq 0 \\x \left (49 A e^{6 i c} + 23 A e^{4 i c} + 7 A e^{2 i c} + A + 15 i B e^{6 i c} + 11 i B e^{4 i c} + 5 i B e^{2 i c} + i B\right ) & \text{otherwise} \end{cases}\right ) e^{- 6 i c}}{8 a^{3}} + \frac{\left (- 3 i A + B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50734, size = 252, normalized size = 1.38 \begin{align*} -\frac{\frac{6 \,{\left (-49 i \, A + 15 \, B\right )} \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{6 \,{\left (i \, A + B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac{96 \,{\left (3 i \, A - B\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac{96 \,{\left (-3 i \, A \tan \left (d x + c\right ) + B \tan \left (d x + c\right ) + A\right )}}{a^{3} \tan \left (d x + c\right )} + \frac{539 \, A \tan \left (d x + c\right )^{3} + 165 i \, B \tan \left (d x + c\right )^{3} - 1821 i \, A \tan \left (d x + c\right )^{2} + 579 \, B \tan \left (d x + c\right )^{2} - 2085 \, A \tan \left (d x + c\right ) - 699 i \, B \tan \left (d x + c\right ) + 819 i \, A - 301 \, B}{a^{3}{\left (i \, \tan \left (d x + c\right ) + 1\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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