Optimal. Leaf size=55 \[ \frac{i (a+i a \tan (c+d x))^5}{5 a^3 d}-\frac{i (a+i a \tan (c+d x))^4}{2 a^2 d} \]
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Rubi [A] time = 0.0431053, 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} \[ \frac{i (a+i a \tan (c+d x))^5}{5 a^3 d}-\frac{i (a+i a \tan (c+d x))^4}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^3-(a+x)^4\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i (a+i a \tan (c+d x))^4}{2 a^2 d}+\frac{i (a+i a \tan (c+d x))^5}{5 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.373512, size = 77, normalized size = 1.4 \[ \frac{a^2 \sec (c) \sec ^5(c+d x) (-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x)+5 i \cos (2 c+d x)+5 \sin (d x)+5 i \cos (d x))}{20 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 85, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{15\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{{\frac{i}{2}}{a}^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12444, size = 76, normalized size = 1.38 \begin{align*} -\frac{6 \, a^{2} \tan \left (d x + c\right )^{5} - 15 i \, a^{2} \tan \left (d x + c\right )^{4} - 30 i \, a^{2} \tan \left (d x + c\right )^{2} - 30 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.18037, size = 329, normalized size = 5.98 \begin{align*} \frac{80 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 80 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 40 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i \, a^{2}}{5 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int - \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 2 i \tan{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1673, size = 76, normalized size = 1.38 \begin{align*} -\frac{2 \, a^{2} \tan \left (d x + c\right )^{5} - 5 i \, a^{2} \tan \left (d x + c\right )^{4} - 10 i \, a^{2} \tan \left (d x + c\right )^{2} - 10 \, a^{2} \tan \left (d x + c\right )}{10 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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