Optimal. Leaf size=104 \[ \frac{64 i a^3 \sec (c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i a^2 \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]
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Rubi [A] time = 0.101283, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3494, 3493} \[ \frac{64 i a^3 \sec (c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i a^2 \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \sec (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac{1}{5} (8 a) \int \sec (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\\ &=\frac{16 i a^2 \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}+\frac{1}{15} \left (32 a^2\right ) \int \sec (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx\\ &=\frac{64 i a^3 \sec (c+d x)}{15 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 i a^2 \sec (c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{2 i a \sec (c+d x) (a+i a \tan (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.314176, size = 93, normalized size = 0.89 \[ \frac{2 a^2 \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (\sin (c-d x)+i \cos (c-d x)) (7 i \sin (2 (c+d x))+23 \cos (2 (c+d x))+20)}{15 d (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.247, size = 90, normalized size = 0.9 \begin{align*}{\frac{2\,{a}^{2} \left ( 32\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+32\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +11\,i\cos \left ( dx+c \right ) -3\,\sin \left ( dx+c \right ) \right ) }{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03447, size = 282, normalized size = 2.71 \begin{align*} \frac{\sqrt{2}{\left (120 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 160 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 64 i \, a^{2}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{15 \,{\left (d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )}} \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{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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