Optimal. Leaf size=191 \[ -\frac{(-3 B+i A) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{(7 A+25 i B) \tan (c+d x)}{8 a^3 d}-\frac{(-3 B+i A) \log (\cos (c+d x))}{a^3 d}-\frac{x (7 A+25 i B)}{8 a^3}+\frac{(-B+i A) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.473484, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {3595, 3525, 3475} \[ -\frac{(-3 B+i A) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{(7 A+25 i B) \tan (c+d x)}{8 a^3 d}-\frac{(-3 B+i A) \log (\cos (c+d x))}{a^3 d}-\frac{x (7 A+25 i B)}{8 a^3}+\frac{(-B+i A) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3595
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac{(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{\int \frac{\tan ^3(c+d x) (4 a (i A-B)+a (A+7 i B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\tan ^2(c+d x) \left (-3 a^2 (5 A+11 i B)+3 a^2 (3 i A-13 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac{(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac{(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\int \tan (c+d x) \left (-48 a^3 (i A-3 B)-6 a^3 (7 A+25 i B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{(7 A+25 i B) x}{8 a^3}+\frac{(7 A+25 i B) \tan (c+d x)}{8 a^3 d}+\frac{(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac{(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{(i A-3 B) \int \tan (c+d x) \, dx}{a^3}\\ &=-\frac{(7 A+25 i B) x}{8 a^3}-\frac{(i A-3 B) \log (\cos (c+d x))}{a^3 d}+\frac{(7 A+25 i B) \tan (c+d x)}{8 a^3 d}+\frac{(i A-B) \tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(5 A+11 i B) \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}-\frac{(i A-3 B) \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 7.01278, size = 1251, normalized size = 6.55 \[ \frac{\sec ^3(c+d x) (-B \cos (3 c-d x)+B \cos (3 c+d x)-i B \sin (3 c-d x)+i B \sin (3 c+d x)) (A+B \tan (c+d x)) (\cos (d x)+i \sin (d x))^3}{2 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) (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) \left (-\frac{1}{2} A \cos ^3(c)-\frac{3}{2} i B \cos ^3(c)-2 i A \sin (c) \cos ^2(c)+6 B \sin (c) \cos ^2(c)+3 A \sin ^2(c) \cos (c)+9 i B \sin ^2(c) \cos (c)+\frac{1}{2} A \cos (c)+\frac{3}{2} i B \cos (c)+2 i A \sin ^3(c)-6 B \sin ^3(c)+i A \sin (c)-3 B \sin (c)-\frac{1}{2} A \sin ^3(c) \tan (c)-\frac{3}{2} i B \sin ^3(c) \tan (c)-\frac{1}{2} A \sin (c) \tan (c)-\frac{3}{2} i B \sin (c) \tan (c)+i (A+3 i B) (\cos (3 c)+i \sin (3 c)) \tan (c)\right ) (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{(11 A+23 i B) \cos (2 d x) \sec ^2(c+d x) \left (\frac{1}{16} i \cos (c)-\frac{\sin (c)}{16}\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{(7 B-5 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{\sec ^2(c+d x) \left (-i A \cos \left (\frac{3 c}{2}\right )+3 B \cos \left (\frac{3 c}{2}\right )+A \sin \left (\frac{3 c}{2}\right )+3 i B \sin \left (\frac{3 c}{2}\right )\right ) \left (\cos \left (\frac{3 c}{2}\right ) \log (\cos (c+d x))+i \sin \left (\frac{3 c}{2}\right ) \log (\cos (c+d x))\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{(A+i B) \cos (6 d x) \sec ^2(c+d x) \left (\frac{1}{48} i \cos (3 c)+\frac{1}{48} \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{(7 A+25 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{(11 A+23 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{(5 A+7 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} \cos (3 c)-\frac{1}{48} i \sin (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 = 0.036, size = 219, normalized size = 1.2 \begin{align*}{\frac{iB\tan \left ( dx+c \right ) }{{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{49\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{16\,{a}^{3}d}}+{\frac{{\frac{15\,i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{{a}^{3}d}}+{\frac{17\,A}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{31\,i}{8}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{7\,i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{9\,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}} \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 = 1.46397, size = 506, normalized size = 2.65 \begin{align*} -\frac{12 \,{\left (15 \, A + 49 i \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (12 \,{\left (15 \, A + 49 i \, B\right )} d x - 66 i \, A + 330 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} -{\left (51 i \, A - 117 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (-13 i \, A + 19 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left ({\left (-96 i \, A + 288 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (-96 i \, A + 288 \, 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 = 35.9561, size = 292, normalized size = 1.53 \begin{align*} - \frac{2 B e^{- 2 i c}}{a^{3} d \left (e^{2 i d x} + e^{- 2 i c}\right )} - \frac{\left (\begin{cases} 15 A x e^{6 i c} - \frac{11 i A e^{4 i c} e^{- 2 i d x}}{2 d} + \frac{5 i A e^{2 i c} e^{- 4 i d x}}{4 d} - \frac{i A e^{- 6 i d x}}{6 d} + 49 i B x e^{6 i c} + \frac{23 B e^{4 i c} e^{- 2 i d x}}{2 d} - \frac{7 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 (15 A e^{6 i c} - 11 A e^{4 i c} + 5 A e^{2 i c} - A + 49 i B e^{6 i c} - 23 i B e^{4 i c} + 7 i B e^{2 i c} - i B\right ) & \text{otherwise} \end{cases}\right ) e^{- 6 i c}}{8 a^{3}} + \frac{\left (- i A + 3 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 = 2.37201, size = 194, normalized size = 1.02 \begin{align*} \frac{\frac{6 \,{\left (i \, A + B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{3}} - \frac{6 \,{\left (-15 i \, A + 49 \, B\right )} \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{96 i \, B \tan \left (d x + c\right )}{a^{3}} - \frac{165 i \, A \tan \left (d x + c\right )^{3} - 539 \, B \tan \left (d x + c\right )^{3} + 291 \, A \tan \left (d x + c\right )^{2} + 1245 i \, B \tan \left (d x + c\right )^{2} - 171 i \, A \tan \left (d x + c\right ) + 981 \, B \tan \left (d x + c\right ) - 29 \, A - 259 i \, B}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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