Optimal. Leaf size=82 \[ \frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]
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Rubi [A] time = 0.0636081, 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} \[ \frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^6 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a-x)^6-4 a (a-x)^7+(a-x)^8\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=\frac{4 i (a-i a \tan (c+d x))^7}{7 a^{11} d}-\frac{i (a-i a \tan (c+d x))^8}{2 a^{12} d}+\frac{i (a-i a \tan (c+d x))^9}{9 a^{13} d}\\ \end{align*}
Mathematica [A] time = 0.559942, size = 136, normalized size = 1.66 \[ \frac{\sec (c) \sec ^9(c+d x) (-63 \sin (2 c+d x)+42 \sin (2 c+3 d x)-42 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x)-63 i \cos (2 c+d x)-42 i \cos (2 c+3 d x)-42 i \cos (4 c+3 d x)+63 \sin (d x)-63 i \cos (d x))}{252 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 99, 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 ) ^{9}}{9}}+{\frac{i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{8}-{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{2\,i}{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}-2\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}-i \left ( \tan \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \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 = 0.984934, size = 131, normalized size = 1.6 \begin{align*} \frac{14 \, \tan \left (d x + c\right )^{9} + 63 i \, \tan \left (d x + c\right )^{8} - 72 \, \tan \left (d x + c\right )^{7} + 84 i \, \tan \left (d x + c\right )^{6} - 252 \, \tan \left (d x + c\right )^{5} - 126 i \, \tan \left (d x + c\right )^{4} - 168 \, \tan \left (d x + c\right )^{3} - 252 i \, \tan \left (d x + c\right )^{2} + 126 \, \tan \left (d x + c\right )}{126 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.8042, size = 494, normalized size = 6.02 \begin{align*} \frac{4608 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1152 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i}{63 \,{\left (a^{4} d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, a^{4} d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, a^{4} d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, a^{4} d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, 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 = 1.19941, size = 131, normalized size = 1.6 \begin{align*} \frac{14 \, \tan \left (d x + c\right )^{9} + 63 i \, \tan \left (d x + c\right )^{8} - 72 \, \tan \left (d x + c\right )^{7} + 84 i \, \tan \left (d x + c\right )^{6} - 252 \, \tan \left (d x + c\right )^{5} - 126 i \, \tan \left (d x + c\right )^{4} - 168 \, \tan \left (d x + c\right )^{3} - 252 i \, \tan \left (d x + c\right )^{2} + 126 \, \tan \left (d x + c\right )}{126 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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