Optimal. Leaf size=165 \[ -\frac{16 i \sqrt{a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}+\frac{32 i \sqrt{e \sec (c+d x)}}{45 a d e^2 \sqrt{a+i a \tan (c+d x)}}+\frac{4 i}{15 a d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac{2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.305762, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3502, 3497, 3488} \[ -\frac{16 i \sqrt{a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}+\frac{32 i \sqrt{e \sec (c+d x)}}{45 a d e^2 \sqrt{a+i a \tan (c+d x)}}+\frac{4 i}{15 a d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}+\frac{2 i}{9 d (a+i a \tan (c+d x))^{3/2} (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}} \, dx &=\frac{2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{2 \int \frac{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx}{3 a}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{8 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{15 a^2}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}+\frac{16 \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{45 a e^2}\\ &=\frac{2 i}{9 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}}+\frac{4 i}{15 a d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{32 i \sqrt{e \sec (c+d x)}}{45 a d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{45 a^2 d (e \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.438154, size = 100, normalized size = 0.61 \[ -\frac{\sec ^3(c+d x) (-54 i \sin (c+d x)+10 i \sin (3 (c+d x))-81 \cos (c+d x)+5 \cos (3 (c+d x)))}{90 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.304, size = 132, normalized size = 0.8 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( 10\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+10\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+6\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +8\,i\cos \left ( dx+c \right ) +16\,\sin \left ( dx+c \right ) \right ) }{45\,{a}^{2}d{e}^{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 = 1.97288, size = 240, normalized size = 1.45 \begin{align*} \frac{5 i \, \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 27 i \, \cos \left (\frac{5}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) - 15 i \, \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 135 i \, \cos \left (\frac{1}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ) + 27 \, \sin \left (\frac{5}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 15 \, \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right ) + 135 \, \sin \left (\frac{1}{9} \, \arctan \left (\sin \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right ), \cos \left (\frac{9}{2} \, d x + \frac{9}{2} \, c\right )\right )\right )}{180 \, a^{\frac{3}{2}} d e^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98072, size = 309, normalized size = 1.87 \begin{align*} \frac{\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}}{\left (-15 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 120 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 162 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 32 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac{9}{2} i \, d x - \frac{9}{2} i \, c\right )}}{180 \, a^{2} d e^{2}} \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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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