Optimal. Leaf size=27 \[ \frac{i (a-i a \tan (c+d x))^5}{5 a^9 d} \]
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Rubi [A] time = 0.0400154, 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} \[ \frac{i (a-i a \tan (c+d x))^5}{5 a^9 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^4 \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=\frac{i (a-i a \tan (c+d x))^5}{5 a^9 d}\\ \end{align*}
Mathematica [B] time = 0.367759, size = 116, normalized size = 4.3 \[ \frac{\sec (c) \sec ^5(c+d x) (-10 \sin (2 c+d x)+5 \sin (2 c+3 d x)-5 \sin (4 c+3 d x)+2 \sin (4 c+5 d x)-10 i \cos (2 c+d x)-5 i \cos (2 c+3 d x)-5 i \cos (4 c+3 d x)+10 \sin (d x)-10 i \cos (d x))}{10 a^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.084, size = 57, normalized size = 2.1 \begin{align*}{\frac{1}{{a}^{4}d} \left ( \tan \left ( dx+c \right ) +{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5}}+i \left ( \tan \left ( dx+c \right ) \right ) ^{4}-2\, \left ( \tan \left ( dx+c \right ) \right ) ^{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 [B] time = 0.977301, size = 77, normalized size = 2.85 \begin{align*} \frac{3 \, \tan \left (d x + c\right )^{5} + 15 i \, \tan \left (d x + c\right )^{4} - 30 \, \tan \left (d x + c\right )^{3} - 30 i \, \tan \left (d x + c\right )^{2} + 15 \, \tan \left (d x + c\right )}{15 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41311, size = 227, normalized size = 8.41 \begin{align*} \frac{32 i}{5 \,{\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{4} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{4} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, 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 [B] time = 1.20862, size = 74, normalized size = 2.74 \begin{align*} \frac{\tan \left (d x + c\right )^{5} + 5 i \, \tan \left (d x + c\right )^{4} - 10 \, \tan \left (d x + c\right )^{3} - 10 i \, \tan \left (d x + c\right )^{2} + 5 \, \tan \left (d x + c\right )}{5 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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