Optimal. Leaf size=31 \[ -\frac{2 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d} \]
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Rubi [A] time = 0.0490911, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {3493} \[ -\frac{2 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3493
Rubi steps
\begin{align*} \int \cos (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=-\frac{2 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.135678, size = 31, normalized size = 1. \[ -\frac{2 i a \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.259, size = 42, normalized size = 1.4 \begin{align*}{\frac{-2\,ia\cos \left ( dx+c \right ) }{d}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.87649, size = 271, normalized size = 8.74 \begin{align*} \frac{2 \,{\left (i \, a^{\frac{3}{2}} - \frac{2 i \, a^{\frac{3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{i \, a^{\frac{3}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (-\frac{2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{\frac{3}{2}}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac{3}{2}}{\left (-\frac{2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20871, size = 105, normalized size = 3.39 \begin{align*} \frac{\sqrt{2}{\left (-i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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