Optimal. Leaf size=27 \[ -\frac{i a^5}{2 d (a-i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.0378988, 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} \[ -\frac{i a^5}{2 d (a-i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^3} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^5}{2 d (a-i a \tan (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.251446, size = 50, normalized size = 1.85 \[ \frac{a^3 (3 \cos (c+d x)-i \sin (c+d x)) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 114, normalized size = 4.2 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{4}}{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}-3\,{a}^{3} \left ( -1/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +1/8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/8\,dx+c/8 \right ) -{\frac{3\,i}{4}}{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{a}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.70067, size = 77, normalized size = 2.85 \begin{align*} \frac{4 i \, a^{3} \tan \left (d x + c\right )^{2} + 8 \, a^{3} \tan \left (d x + c\right ) - 4 i \, a^{3}}{8 \,{\left (\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16282, size = 89, normalized size = 3.3 \begin{align*} \frac{-i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.481706, size = 82, normalized size = 3.04 \begin{align*} \begin{cases} \frac{- 4 i a^{3} d e^{4 i c} e^{4 i d x} - 8 i a^{3} d e^{2 i c} e^{2 i d x}}{32 d^{2}} & \text{for}\: 32 d^{2} \neq 0 \\x \left (\frac{a^{3} e^{4 i c}}{2} + \frac{a^{3} e^{2 i c}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29083, size = 182, normalized size = 6.74 \begin{align*} \frac{-8 i \, a^{3} e^{\left (12 i \, d x + 8 i \, c\right )} - 48 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} - 112 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} - 128 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} - 16 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} - 72 i \, a^{3} e^{\left (4 i \, d x\right )}}{64 \,{\left (d e^{\left (8 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + 4 \, d e^{\left (2 i \, d x - 2 i \, c\right )} + 6 \, d e^{\left (4 i \, d x\right )} + d e^{\left (-4 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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