Optimal. Leaf size=132 \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{5 d}-\frac{16 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{15 d}+\frac{8 i a \sec ^2(c+d x) (e \cos (c+d x))^{5/2}}{15 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.288011, antiderivative size = 132, 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, 3497, 3502, 3488} \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{5 d}-\frac{16 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{15 d}+\frac{8 i a \sec ^2(c+d x) (e \cos (c+d x))^{5/2}}{15 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3497
Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)} \, dx &=\left ((e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx\\ &=-\frac{2 i (e \cos (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{\left (4 a (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx}{5 e^2}\\ &=\frac{8 i a (e \cos (c+d x))^{5/2} \sec ^2(c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i (e \cos (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{\left (8 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{15 e^2}\\ &=\frac{8 i a (e \cos (c+d x))^{5/2} \sec ^2(c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i (e \cos (c+d x))^{5/2} \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{16 i (e \cos (c+d x))^{5/2} \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}\\ \end{align*}
Mathematica [A] time = 0.336596, size = 63, normalized size = 0.48 \[ \frac{i e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)} (-4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.373, size = 80, normalized size = 0.6 \begin{align*}{\frac{2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -16\,i}{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{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 = 3.11623, size = 200, normalized size = 1.52 \begin{align*} \frac{{\left (5 i \, e^{2} \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) - 3 i \, e^{2} \cos \left (\frac{5}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) - 30 i \, e^{2} \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) + 5 \, e^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, e^{2} \sin \left (\frac{5}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) + 30 \, e^{2} \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right )\right )} \sqrt{a} \sqrt{e}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06499, size = 255, normalized size = 1.93 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{1}{2}}{\left (-3 i \, e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 30 i \, e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i \, e^{2}\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{3}{2} i \, d x - \frac{3}{2} i \, c\right )}}{30 \, 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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