Optimal. Leaf size=128 \[ -\frac{3 i}{16 a^4 d (1+i \tan (c+d x))}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac{x}{16 a^4}+\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.178446, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3546, 3540, 3526, 8} \[ -\frac{3 i}{16 a^4 d (1+i \tan (c+d x))}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac{x}{16 a^4}+\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3546
Rule 3540
Rule 3526
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}-\frac{i \int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{2 a}\\ &=\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}-\frac{\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{4 a^2}\\ &=\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac{\int \frac{a-2 i a \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{8 a^4}\\ &=-\frac{3 i}{16 a^4 d (1+i \tan (c+d x))}+\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac{\int 1 \, dx}{16 a^4}\\ &=\frac{x}{16 a^4}-\frac{3 i}{16 a^4 d (1+i \tan (c+d x))}+\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}\\ \end{align*}
Mathematica [A] time = 0.211376, size = 98, normalized size = 0.77 \[ \frac{\sec ^4(c+d x) (32 \sin (2 (c+d x))+24 i d x \sin (4 (c+d x))+3 \sin (4 (c+d x))-64 i \cos (2 (c+d x))+3 (8 d x+i) \cos (4 (c+d x))+36 i)}{384 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 118, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{8}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{{\frac{i}{32}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{4}}}-{\frac{{\frac{17\,i}{16}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{7}{12\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{15}{16\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{i}{32}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{4}}} \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 = 2.59287, size = 205, normalized size = 1.6 \begin{align*} \frac{{\left (24 \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 48 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 36 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 16 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\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 = 2.33076, size = 190, normalized size = 1.48 \begin{align*} \begin{cases} \frac{\left (- 98304 i a^{12} d^{3} e^{18 i c} e^{- 2 i d x} + 73728 i a^{12} d^{3} e^{16 i c} e^{- 4 i d x} - 32768 i a^{12} d^{3} e^{14 i c} e^{- 6 i d x} + 6144 i a^{12} d^{3} e^{12 i c} e^{- 8 i d x}\right ) e^{- 20 i c}}{786432 a^{16} d^{4}} & \text{for}\: 786432 a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (\frac{\left (e^{8 i c} - 4 e^{6 i c} + 6 e^{4 i c} - 4 e^{2 i c} + 1\right ) e^{- 8 i c}}{16 a^{4}} - \frac{1}{16 a^{4}}\right ) & \text{otherwise} \end{cases} + \frac{x}{16 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.51677, size = 124, normalized size = 0.97 \begin{align*} -\frac{-\frac{12 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} + \frac{12 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} + \frac{-25 i \, \tan \left (d x + c\right )^{4} + 260 \, \tan \left (d x + c\right )^{3} - 522 i \, \tan \left (d x + c\right )^{2} - 388 \, \tan \left (d x + c\right ) + 103 i}{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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