Optimal. Leaf size=85 \[ \frac{4 i a e \sec (c+d x) \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.212166, antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3515, 3497, 3488} \[ \frac{4 i a \sec ^2(c+d x) (e \cos (c+d x))^{3/2}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)} \, dx &=\left ((e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx\\ &=-\frac{2 i (e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{\left (2 a (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{3 e^2}\\ &=\frac{4 i a (e \cos (c+d x))^{3/2} \sec ^2(c+d x)}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i (e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.203282, size = 56, normalized size = 0.66 \[ \frac{2 e \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)} (2 \sin (c+d x)+i \cos (c+d x))}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.338, size = 70, normalized size = 0.8 \begin{align*}{\frac{2\,i\cos \left ( dx+c \right ) +4\,\sin \left ( dx+c \right ) }{3\,d\cos \left ( dx+c \right ) } \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\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 [A] time = 3.02176, size = 80, normalized size = 0.94 \begin{align*} \frac{{\left (-i \, e \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 i \, e \cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + e \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, e \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a} \sqrt{e}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06136, size = 204, normalized size = 2.4 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{1}{2}} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e}{\left (-i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \, 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(-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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