Optimal. Leaf size=27 \[ \frac{i (a-i a \tan (c+d x))^3}{3 a^5 d} \]
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Rubi [A] time = 0.0444558, 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} \[ \frac{i (a-i a \tan (c+d x))^3}{3 a^5 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=\frac{i (a-i a \tan (c+d x))^3}{3 a^5 d}\\ \end{align*}
Mathematica [B] time = 0.209109, size = 68, normalized size = 2.52 \[ \frac{\sec (c) \sec ^3(c+d x) (-3 \sin (2 c+d x)+2 \sin (2 c+3 d x)-3 i \cos (2 c+d x)+3 \sin (d x)-3 i \cos (d x))}{6 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 36, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{2}d} \left ( \tan \left ( dx+c \right ) -{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-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 = 1.09483, size = 47, normalized size = 1.74 \begin{align*} -\frac{\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9394, size = 139, normalized size = 5.15 \begin{align*} \frac{8 i}{3 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 = 1.18927, size = 47, normalized size = 1.74 \begin{align*} -\frac{\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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