Optimal. Leaf size=27 \[ \frac{i (a-i a \tan (c+d x))^4}{4 a^7 d} \]
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Rubi [A] time = 0.039479, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 32} \[ \frac{i (a-i a \tan (c+d x))^4}{4 a^7 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac{i (a-i a \tan (c+d x))^4}{4 a^7 d}\\ \end{align*}
Mathematica [B] time = 0.378706, size = 84, normalized size = 3.11 \[ \frac{\sec (c) \sec ^4(c+d x) (2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x)-2 i \cos (c+2 d x)-2 i \cos (3 c+2 d x)-3 \sin (c)-3 i \cos (c))}{4 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 47, normalized size = 1.7 \begin{align*}{\frac{\tan \left ( dx+c \right ) +{\frac{i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}- \left ( \tan \left ( dx+c \right ) \right ) ^{3}-{\frac{3\,i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.970975, size = 63, normalized size = 2.33 \begin{align*} -\frac{-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30315, size = 177, normalized size = 6.56 \begin{align*} \frac{4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20958, size = 63, normalized size = 2.33 \begin{align*} -\frac{-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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