Optimal. Leaf size=80 \[ \frac{i (a-i a \tan (c+d x))^6}{6 a^7 d}-\frac{4 i (a-i a \tan (c+d x))^5}{5 a^6 d}+\frac{i (a-i a \tan (c+d x))^4}{a^5 d} \]
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Rubi [A] time = 0.0620516, antiderivative size = 80, 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} \[ \frac{i (a-i a \tan (c+d x))^6}{6 a^7 d}-\frac{4 i (a-i a \tan (c+d x))^5}{5 a^6 d}+\frac{i (a-i a \tan (c+d x))^4}{a^5 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a-x)^3-4 a (a-x)^4+(a-x)^5\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac{i (a-i a \tan (c+d x))^4}{a^5 d}-\frac{4 i (a-i a \tan (c+d x))^5}{5 a^6 d}+\frac{i (a-i a \tan (c+d x))^6}{6 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.258387, size = 60, normalized size = 0.75 \[ \frac{\sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)+6 \sin (3 c+4 d x)+\sin (5 c+6 d x)-10 \sin (c)-10 i \cos (c))}{60 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 68, normalized size = 0.9 \begin{align*}{\frac{1}{ad} \left ( \tan \left ( dx+c \right ) -{\frac{i}{6}} \left ( \tan \left ( dx+c \right ) \right ) ^{6}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5}}-{\frac{i}{2}} \left ( \tan \left ( dx+c \right ) \right ) ^{4}+{\frac{2\, \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 = 1.13411, size = 90, normalized size = 1.12 \begin{align*} -\frac{10 i \, \tan \left (d x + c\right )^{6} - 12 \, \tan \left (d x + c\right )^{5} + 30 i \, \tan \left (d x + c\right )^{4} - 40 \, \tan \left (d x + c\right )^{3} + 30 i \, \tan \left (d x + c\right )^{2} - 60 \, \tan \left (d x + c\right )}{60 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86957, size = 333, normalized size = 4.16 \begin{align*} \frac{240 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 96 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 16 i}{15 \,{\left (a d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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 = 1.15693, size = 90, normalized size = 1.12 \begin{align*} -\frac{5 i \, \tan \left (d x + c\right )^{6} - 6 \, \tan \left (d x + c\right )^{5} + 15 i \, \tan \left (d x + c\right )^{4} - 20 \, \tan \left (d x + c\right )^{3} + 15 i \, \tan \left (d x + c\right )^{2} - 30 \, \tan \left (d x + c\right )}{30 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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