Optimal. Leaf size=80 \[ \frac{4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.165989, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3502, 3488} \[ \frac{4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 \int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^{3/2}} \, dx}{7 a}\\ &=\frac{2 i (e \sec (c+d x))^{3/2}}{7 d (a+i a \tan (c+d x))^{5/2}}+\frac{4 i (e \sec (c+d x))^{3/2}}{21 a d (a+i a \tan (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.199161, size = 63, normalized size = 0.79 \[ \frac{2 (2 \tan (c+d x)-5 i) (e \sec (c+d x))^{3/2}}{21 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.291, size = 112, normalized size = 1.4 \begin{align*}{\frac{-{\frac{2\,i}{21}} \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( 12\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -12\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+i\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) }{d{a}^{3}} \left ({\frac{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 = 1.9025, size = 116, normalized size = 1.45 \begin{align*} \frac{{\left (3 i \, e \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 7 i \, e \cos \left (\frac{3}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right ) + 3 \, e \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 7 \, e \sin \left (\frac{3}{7} \, \arctan \left (\sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ), \cos \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right )\right )\right )\right )} \sqrt{e}}{21 \, a^{\frac{5}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01614, size = 232, normalized size = 2.9 \begin{align*} \frac{{\left (7 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 10 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}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac{7}{2} i \, d x - \frac{7}{2} i \, c\right )}}{21 \, a^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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