Optimal. Leaf size=27 \[ -\frac{i (a+i a \tan (c+d x))^4}{4 a d} \]
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Rubi [A] time = 0.0365777, 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} \[ -\frac{i (a+i a \tan (c+d x))^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{i (a+i a \tan (c+d x))^4}{4 a d}\\ \end{align*}
Mathematica [B] time = 0.488852, size = 84, normalized size = 3.11 \[ \frac{a^3 \sec (c) \sec ^4(c+d x) (2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x)+2 i \cos (c+2 d x)+2 i \cos (3 c+2 d x)-3 \sin (c)+3 i \cos (c))}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 73, normalized size = 2.7 \begin{align*}{\frac{1}{d} \left ({\frac{-{\frac{i}{4}}{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{\frac{3\,i}{2}}{a}^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{3}\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 = 1.11796, size = 28, normalized size = 1.04 \begin{align*} -\frac{i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{4 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.19093, size = 284, normalized size = 10.52 \begin{align*} \frac{16 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 24 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 16 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a^{3}}{d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \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^{3} \left (\int - 3 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 i \tan{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - i \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22505, size = 76, normalized size = 2.81 \begin{align*} -\frac{i \, a^{3} \tan \left (d x + c\right )^{4} + 4 \, a^{3} \tan \left (d x + c\right )^{3} - 6 i \, a^{3} \tan \left (d x + c\right )^{2} - 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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