Optimal. Leaf size=55 \[ \frac{i (a-i a \tan (c+d x))^6}{3 a^{10} d}-\frac{i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]
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Rubi [A] time = 0.0472249, 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} \[ \frac{i (a-i a \tan (c+d x))^6}{3 a^{10} d}-\frac{i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^5 (a+x) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a-x)^5-(a-x)^6\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d}\\ &=\frac{i (a-i a \tan (c+d x))^6}{3 a^{10} d}-\frac{i (a-i a \tan (c+d x))^7}{7 a^{11} d}\\ \end{align*}
Mathematica [B] time = 0.347308, size = 127, normalized size = 2.31 \[ \frac{\sec (c) \sec ^7(c+d x) (-35 \sin (2 c+d x)+21 \sin (2 c+3 d x)-21 \sin (4 c+3 d x)+14 \sin (4 c+5 d x)+2 \sin (6 c+7 d x)-35 i \cos (2 c+d x)-21 i \cos (2 c+3 d x)-21 i \cos (4 c+3 d x)+35 \sin (d x)-35 i \cos (d x))}{84 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 67, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{4}d} \left ( \tan \left ( dx+c \right ) +{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{2\,i}{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}- \left ( \tan \left ( dx+c \right ) \right ) ^{5}-{\frac{5\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3}}-2\,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 = 0.98677, size = 90, normalized size = 1.64 \begin{align*} \frac{3 \, \tan \left (d x + c\right )^{7} + 14 i \, \tan \left (d x + c\right )^{6} - 21 \, \tan \left (d x + c\right )^{5} - 35 \, \tan \left (d x + c\right )^{3} - 42 i \, \tan \left (d x + c\right )^{2} + 21 \, \tan \left (d x + c\right )}{21 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.50114, size = 360, normalized size = 6.55 \begin{align*} \frac{448 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 64 i}{21 \,{\left (a^{4} d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, a^{4} d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} 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 = 1.16022, size = 90, normalized size = 1.64 \begin{align*} \frac{3 \, \tan \left (d x + c\right )^{7} + 14 i \, \tan \left (d x + c\right )^{6} - 21 \, \tan \left (d x + c\right )^{5} - 35 \, \tan \left (d x + c\right )^{3} - 42 i \, \tan \left (d x + c\right )^{2} + 21 \, \tan \left (d x + c\right )}{21 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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