Optimal. Leaf size=116 \[ \frac{i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac{x}{16 a^4}+\frac{i}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{8 d (a+i a \tan (c+d x))^4} \]
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Rubi [A] time = 0.0690521, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3479, 8} \[ \frac{i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac{x}{16 a^4}+\frac{i}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{8 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (c+d x))^4} \, dx &=\frac{i}{8 d (a+i a \tan (c+d x))^4}+\frac{\int \frac{1}{(a+i a \tan (c+d x))^3} \, dx}{2 a}\\ &=\frac{i}{8 d (a+i a \tan (c+d x))^4}+\frac{i}{12 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{1}{(a+i a \tan (c+d x))^2} \, dx}{4 a^2}\\ &=\frac{i}{8 d (a+i a \tan (c+d x))^4}+\frac{i}{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{1}{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=\frac{i}{8 d (a+i a \tan (c+d x))^4}+\frac{i}{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{i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int 1 \, dx}{16 a^4}\\ &=\frac{x}{16 a^4}+\frac{i}{8 d (a+i a \tan (c+d x))^4}+\frac{i}{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{i}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.209361, size = 98, normalized size = 0.84 \[ \frac{\sec ^4(c+d x) (-32 \sin (2 (c+d x))+24 i d x \sin (4 (c+d x))+3 \sin (4 (c+d x))+64 i \cos (2 (c+d x))+3 (8 d x+i) \cos (4 (c+d x))+36 i)}{384 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 118, normalized size = 1. \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{i}{16}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{1}{12\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{1}{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 = 2.18034, size = 205, normalized size = 1.77 \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 = 1.67513, size = 190, normalized size = 1.64 \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 = 1.26729, size = 124, normalized size = 1.07 \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} - 124 \, \tan \left (d x + c\right )^{3} + 246 i \, \tan \left (d x + c\right )^{2} + 252 \, \tan \left (d x + c\right ) - 153 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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