Optimal. Leaf size=159 \[ \frac{5 A-29 i B}{48 a^4 d (1+i \tan (c+d x))}-\frac{A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{x (B+i A)}{16 a^4}+\frac{(-B+i A) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.465667, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3595, 3590, 3526, 8} \[ \frac{5 A-29 i B}{48 a^4 d (1+i \tan (c+d x))}-\frac{A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{x (B+i A)}{16 a^4}+\frac{(-B+i A) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3595
Rule 3590
Rule 3526
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx &=\frac{(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}-\frac{\int \frac{\tan ^2(c+d x) (3 a (i A-B)-a (A-7 i B) \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2}\\ &=\frac{(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\tan (c+d x) \left (-4 a^2 (A+5 i B)-8 a^2 (i A+4 B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4}\\ &=-\frac{A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}-\frac{i \int \frac{4 a^3 (A-13 i B)-16 a^3 (i A+4 B) \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{96 a^6}\\ &=-\frac{A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{5 A-29 i B}{48 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{(i A+B) \int 1 \, dx}{16 a^4}\\ &=\frac{(i A+B) x}{16 a^4}-\frac{A-13 i B}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(i A-B) \tan ^3(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(A+5 i B) \tan ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{5 A-29 i B}{48 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.35887, size = 158, normalized size = 0.99 \[ \frac{\sec ^4(c+d x) (16 (A-4 i B) \cos (2 (c+d x))+3 (8 i A d x+A+8 B d x+i B) \cos (4 (c+d x))+32 i A \sin (2 (c+d x))-3 i A \sin (4 (c+d x))-24 A d x \sin (4 (c+d x))+32 B \sin (2 (c+d x))+24 i B d x \sin (4 (c+d x))+3 B \sin (4 (c+d x))+36 i B)}{384 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 244, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{32\,{a}^{4}d}}-{\frac{{\frac{i}{32}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{{a}^{4}d}}-{\frac{{\frac{5\,i}{12}}A}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{7\,B}{12\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{{\frac{i}{16}}A}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{15\,B}{16\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{A}{8\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}+{\frac{{\frac{i}{8}}B}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{7\,A}{16\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{{\frac{17\,i}{16}}B}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{32\,{a}^{4}d}}+{\frac{{\frac{i}{32}}B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{4}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 = 1.43188, size = 262, normalized size = 1.65 \begin{align*} \frac{{\left ({\left (24 i \, A + 24 \, B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} + 24 \,{\left (A - 2 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 36 i \, B e^{\left (4 i \, d x + 4 i \, c\right )} - 8 \,{\left (A + 2 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, A + 3 i \, B\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.53229, size = 303, normalized size = 1.91 \begin{align*} \begin{cases} \frac{\left (294912 i B a^{12} d^{3} e^{16 i c} e^{- 4 i d x} + \left (24576 A a^{12} d^{3} e^{12 i c} + 24576 i B a^{12} d^{3} e^{12 i c}\right ) e^{- 8 i d x} + \left (- 65536 A a^{12} d^{3} e^{14 i c} - 131072 i B a^{12} d^{3} e^{14 i c}\right ) e^{- 6 i d x} + \left (196608 A a^{12} d^{3} e^{18 i c} - 393216 i B a^{12} d^{3} e^{18 i c}\right ) e^{- 2 i d x}\right ) e^{- 20 i c}}{3145728 a^{16} d^{4}} & \text{for}\: 3145728 a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (- \frac{i A + B}{16 a^{4}} + \frac{\left (i A e^{8 i c} - 2 i A e^{6 i c} + 2 i A e^{2 i c} - i A + B e^{8 i c} - 4 B e^{6 i c} + 6 B e^{4 i c} - 4 B e^{2 i c} + B\right ) e^{- 8 i c}}{16 a^{4}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (i A + B\right )}{16 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.2655, size = 207, normalized size = 1.3 \begin{align*} -\frac{\frac{12 \,{\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} - \frac{12 \,{\left (A - i \, B\right )} \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} + \frac{25 \, A \tan \left (d x + c\right )^{4} - 25 i \, B \tan \left (d x + c\right )^{4} - 124 i \, A \tan \left (d x + c\right )^{3} + 260 \, B \tan \left (d x + c\right )^{3} - 54 \, A \tan \left (d x + c\right )^{2} - 522 i \, B \tan \left (d x + c\right )^{2} - 4 i \, A \tan \left (d x + c\right ) - 388 \, B \tan \left (d x + c\right ) - 7 \, A + 103 i \, B}{a^{4}{\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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