Optimal. Leaf size=126 \[ -\frac{8 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{15 a d}+\frac{2 i (e \cos (c+d x))^{3/2}}{5 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i \sec ^2(c+d x) (e \cos (c+d x))^{3/2}}{15 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.312992, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3515, 3502, 3497, 3488} \[ -\frac{8 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{15 a d}+\frac{2 i (e \cos (c+d x))^{3/2}}{5 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i \sec ^2(c+d x) (e \cos (c+d x))^{3/2}}{15 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3502
Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{(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{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{2 i (e \cos (c+d x))^{3/2}}{5 d \sqrt{a+i a \tan (c+d x)}}+\frac{\left (4 (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}{5 a}\\ &=\frac{2 i (e \cos (c+d x))^{3/2}}{5 d \sqrt{a+i a \tan (c+d x)}}-\frac{8 i (e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{15 a d}+\frac{\left (8 (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}{15 e^2}\\ &=\frac{2 i (e \cos (c+d x))^{3/2}}{5 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i (e \cos (c+d x))^{3/2} \sec ^2(c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}-\frac{8 i (e \cos (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}{15 a d}\\ \end{align*}
Mathematica [A] time = 0.335607, size = 63, normalized size = 0.5 \[ -\frac{i e^2 (4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.35, size = 100, normalized size = 0.8 \begin{align*}{\frac{6\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+6\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +8\,i\cos \left ( dx+c \right ) +16\,\sin \left ( dx+c \right ) }{15\,ad\cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.56851, size = 184, normalized size = 1.46 \begin{align*} \frac{{\left (3 i \, e \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) - 5 i \, e \cos \left (\frac{3}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + 30 i \, e \cos \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + 3 \, e \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, e \sin \left (\frac{3}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + 30 \, e \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right )\right )} \sqrt{e}}{30 \, \sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0634, size = 252, normalized size = 2. \begin{align*} \frac{\sqrt{2} \sqrt{\frac{1}{2}}{\left (-5 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 30 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt{e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac{5}{2} i \, d x - \frac{5}{2} i \, c\right )}}{30 \, a 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 \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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