Optimal. Leaf size=65 \[ \frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}-\frac{8 i a^2 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d} \]
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Rubi [A] time = 0.0966806, antiderivative size = 65, 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 = {3494, 3493} \[ \frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}-\frac{8 i a^2 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \cos (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}+(4 a) \int \cos (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac{8 i a^2 \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 i a \cos (c+d x) (a+i a \tan (c+d x))^{3/2}}{d}\\ \end{align*}
Mathematica [A] time = 0.253252, size = 46, normalized size = 0.71 \[ -\frac{2 i a^2 \sqrt{a+i a \tan (c+d x)} (3 \cos (c+d x)-i \sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.279, size = 53, normalized size = 0.8 \begin{align*} -2\,{\frac{{a}^{2} \left ( 3\,i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) \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 = 1.93972, size = 447, normalized size = 6.88 \begin{align*} \frac{2 \,{\left (-3 i \, a^{\frac{5}{2}} - \frac{2 \, a^{\frac{5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{9 i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{4 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{9 i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{2 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{3 i \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\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{4 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97035, size = 116, normalized size = 1.78 \begin{align*} \frac{\sqrt{2}{\left (-2 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, a^{2}\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{5}{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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