Optimal. Leaf size=55 \[ \frac{2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac{i (a-i a \tan (c+d x))^4}{4 a^5 d} \]
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Rubi [A] time = 0.0511579, 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} \[ \frac{2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac{i (a-i a \tan (c+d x))^4}{4 a^5 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a-x)^2-(a-x)^3\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=\frac{2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac{i (a-i a \tan (c+d x))^4}{4 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.166073, size = 49, normalized size = 0.89 \[ \frac{\sec (c) \sec ^4(c+d x) (4 \sin (c+2 d x)+\sin (3 c+4 d x)-3 \sin (c)-3 i \cos (c))}{12 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 47, normalized size = 0.9 \begin{align*}{\frac{1}{ad} \left ( \tan \left ( dx+c \right ) -{\frac{i}{4}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-{\frac{i}{2}} \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 = 1.39724, size = 63, normalized size = 1.15 \begin{align*} \frac{-3 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} + 12 \, \tan \left (d x + c\right )}{12 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95761, size = 208, normalized size = 3.78 \begin{align*} \frac{16 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i}{3 \,{\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a 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 = 1.19848, size = 63, normalized size = 1.15 \begin{align*} -\frac{3 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} - 12 \, \tan \left (d x + c\right )}{12 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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