Optimal. Leaf size=55 \[ \frac{i (a-i a \tan (c+d x))^4}{2 a^6 d}-\frac{i (a-i a \tan (c+d x))^5}{5 a^7 d} \]
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Rubi [A] time = 0.0505928, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac{i (a-i a \tan (c+d x))^4}{2 a^6 d}-\frac{i (a-i a \tan (c+d x))^5}{5 a^7 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a-x)^3-(a-x)^4\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac{i (a-i a \tan (c+d x))^4}{2 a^6 d}-\frac{i (a-i a \tan (c+d x))^5}{5 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.328308, size = 77, normalized size = 1.4 \[ \frac{\sec (c) \sec ^5(c+d x) (-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x)-5 i \cos (2 c+d x)+5 \sin (d x)-5 i \cos (d x))}{20 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 47, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}d} \left ( \tan \left ( dx+c \right ) -{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5}}-{\frac{i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}-i \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15832, size = 63, normalized size = 1.15 \begin{align*} -\frac{2 \, \tan \left (d x + c\right )^{5} + 5 i \, \tan \left (d x + c\right )^{4} + 10 i \, \tan \left (d x + c\right )^{2} - 10 \, \tan \left (d x + c\right )}{10 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12755, size = 267, normalized size = 4.85 \begin{align*} \frac{40 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i}{5 \,{\left (a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \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 [A] time = 1.15277, size = 63, normalized size = 1.15 \begin{align*} -\frac{2 \, \tan \left (d x + c\right )^{5} + 5 i \, \tan \left (d x + c\right )^{4} + 10 i \, \tan \left (d x + c\right )^{2} - 10 \, \tan \left (d x + c\right )}{10 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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