Optimal. Leaf size=179 \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{7 d}+\frac{32 i a \sec ^4(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{35 d}+\frac{12 i a \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.382654, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3515, 3497, 3502, 3488} \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{7 d}+\frac{32 i a \sec ^4(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{35 d}+\frac{12 i a \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt{a+i a \tan (c+d x)}} \]
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))^{7/2} \sqrt{a+i a \tan (c+d x)} \, dx &=\left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx\\ &=-\frac{2 i (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{\left (6 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx}{7 e^2}\\ &=\frac{12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{\left (24 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{35 e^2}\\ &=\frac{12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{\left (16 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{35 e^4}\\ &=\frac{12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt{a+i a \tan (c+d x)}}+\frac{32 i a (e \cos (c+d x))^{7/2} \sec ^4(c+d x)}{35 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}\\ \end{align*}
Mathematica [A] time = 0.536439, size = 80, normalized size = 0.45 \[ \frac{e^3 \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)} (70 \sin (c+d x)+6 \sin (3 (c+d x))+35 i \cos (c+d x)+i \cos (3 (c+d x)))}{70 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.388, size = 97, normalized size = 0.5 \begin{align*}{\frac{2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+12\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +16\,i\cos \left ( dx+c \right ) +32\,\sin \left ( dx+c \right ) }{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}} \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{7}{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 = 2.92401, size = 273, normalized size = 1.53 \begin{align*} \frac{{\left (7 i \, e^{3} \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) - 5 i \, e^{3} \cos \left (\frac{7}{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 ) - 35 i \, e^{3} \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 ) + 105 i \, e^{3} \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 ) + 7 \, e^{3} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, e^{3} \sin \left (\frac{7}{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 ) + 35 \, e^{3} \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 ) + 105 \, e^{3} \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{a} \sqrt{e}}{140 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07285, size = 300, normalized size = 1.68 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{1}{2}}{\left (-5 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 35 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 105 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{3}\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 )}}{140 \, 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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