Optimal. Leaf size=82 \[ \frac{i (a-i a \tan (c+d x))^7}{7 a^9 d}-\frac{2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac{4 i (a-i a \tan (c+d x))^5}{5 a^7 d} \]
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Rubi [A] time = 0.0625975, antiderivative size = 82, 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} \[ \frac{i (a-i a \tan (c+d x))^7}{7 a^9 d}-\frac{2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac{4 i (a-i a \tan (c+d x))^5}{5 a^7 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^4 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a-x)^4-4 a (a-x)^5+(a-x)^6\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=\frac{4 i (a-i a \tan (c+d x))^5}{5 a^7 d}-\frac{2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac{i (a-i a \tan (c+d x))^7}{7 a^9 d}\\ \end{align*}
Mathematica [A] time = 0.418424, size = 90, normalized size = 1.1 \[ \frac{\sec (c) \sec ^7(c+d x) (-35 \sin (2 c+d x)+42 \sin (2 c+3 d x)+14 \sin (4 c+5 d x)+2 \sin (6 c+7 d x)-35 i \cos (2 c+d x)+35 \sin (d x)-35 i \cos (d x))}{210 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 78, normalized size = 1. \begin{align*}{\frac{1}{{a}^{2}d} \left ( \tan \left ( dx+c \right ) -{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{i}{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5}}-i \left ( \tan \left ( dx+c \right ) \right ) ^{4}+{\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 = 1.13657, size = 104, normalized size = 1.27 \begin{align*} -\frac{15 \, \tan \left (d x + c\right )^{7} + 35 i \, \tan \left (d x + c\right )^{6} + 21 \, \tan \left (d x + c\right )^{5} + 105 i \, \tan \left (d x + c\right )^{4} - 35 \, \tan \left (d x + c\right )^{3} + 105 i \, \tan \left (d x + c\right )^{2} - 105 \, \tan \left (d x + c\right )}{105 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11047, size = 402, normalized size = 4.9 \begin{align*} \frac{2688 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 896 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i}{105 \,{\left (a^{2} d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, a^{2} d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, 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 = 1.16977, size = 104, normalized size = 1.27 \begin{align*} -\frac{15 \, \tan \left (d x + c\right )^{7} + 35 i \, \tan \left (d x + c\right )^{6} + 21 \, \tan \left (d x + c\right )^{5} + 105 i \, \tan \left (d x + c\right )^{4} - 35 \, \tan \left (d x + c\right )^{3} + 105 i \, \tan \left (d x + c\right )^{2} - 105 \, \tan \left (d x + c\right )}{105 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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