Optimal. Leaf size=46 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0323892, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3486, 2633} \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 2633
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos ^3(c+d x)}{3 d}+a \int \cos ^3(c+d x) \, dx\\ &=-\frac{i a \cos ^3(c+d x)}{3 d}-\frac{a \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{i a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0095743, size = 46, normalized size = 1. \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 37, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{3}}a \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{a \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12166, size = 49, normalized size = 1.07 \begin{align*} -\frac{i \, a \cos \left (d x + c\right )^{3} +{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10006, size = 119, normalized size = 2.59 \begin{align*} \frac{{\left (-i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 6 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, a\right )} e^{\left (-i \, d x - i \, c\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.558693, size = 107, normalized size = 2.33 \begin{align*} \begin{cases} \frac{\left (- 8 i a d^{2} e^{4 i c} e^{3 i d x} - 48 i a d^{2} e^{2 i c} e^{i d x} + 24 i a d^{2} e^{- i d x}\right ) e^{- i c}}{96 d^{3}} & \text{for}\: 96 d^{3} e^{i c} \neq 0 \\\frac{x \left (a e^{4 i c} + 2 a e^{2 i c} + a\right ) e^{- i c}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14382, size = 265, normalized size = 5.76 \begin{align*} -\frac{{\left (9 \, a e^{\left (i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 6 \, a e^{\left (i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 9 \, a e^{\left (i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 6 \, a e^{\left (i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 3 \, a e^{\left (i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 3 \, a e^{\left (i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 4 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i \, a\right )} e^{\left (-i \, d x - i \, c\right )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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