Optimal. Leaf size=81 \[ -\frac{4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}} \]
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Rubi [A] time = 0.153757, antiderivative size = 81, 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 = {3497, 3488} \[ -\frac{4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{7/2}} \, dx &=-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}}+\frac{(2 a) \int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2}\\ &=-\frac{4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.39799, size = 92, normalized size = 1.14 \[ -\frac{2 a^2 (2 \tan (c+d x)+5 i) \sqrt{a+i a \tan (c+d x)} (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{21 d e^2 (\cos (d x)+i \sin (d x))^2 (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.305, size = 105, normalized size = 1.3 \begin{align*} -{\frac{2\,{a}^{2} \left ( 6\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}-6\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -i\cos \left ( dx+c \right ) -2\,\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{21\,d{e}^{7}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.07442, size = 127, normalized size = 1.57 \begin{align*} \frac{{\left (-7 i \, a^{2} \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) - 3 i \, a^{2} \cos \left (\frac{7}{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 ) + 7 \, a^{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, a^{2} \sin \left (\frac{7}{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}}{21 \, d e^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32796, size = 240, normalized size = 2.96 \begin{align*} \frac{{\left (-3 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 10 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 7 i \, a^{2}\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{3}{2} i \, d x + \frac{3}{2} i \, c\right )}}{21 \, d e^{4}} \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 (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\left (e \sec \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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