Optimal. Leaf size=35 \[ -\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.0598799, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {3493} \[ -\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3493
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.323248, size = 69, normalized size = 1.97 \[ \frac{2 a^2 \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (c+3 d x)-i \cos (c+3 d x))}{3 d (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.293, size = 63, normalized size = 1.8 \begin{align*} -{\frac{2\,{a}^{2} \left ( i\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,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 = 2.14478, size = 443, normalized size = 12.66 \begin{align*} \frac{2 \,{\left (i \, a^{\frac{5}{2}} - \frac{4 i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\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{5}{2}}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{2}}{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}^{\frac{5}{2}}{\left (-\frac{6 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{18 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{18 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{6 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{6 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{3 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04885, size = 157, normalized size = 4.49 \begin{align*} \frac{\sqrt{2}{\left (-i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{6 \, 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{5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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