Optimal. Leaf size=36 \[ \frac{2 i}{d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 0.139454, antiderivative size = 36, 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 = {3515, 3488} \[ \frac{2 i}{d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3488
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{\int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{\sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 i}{d \sqrt{e \cos (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.172804, size = 36, normalized size = 1. \[ \frac{2 i}{d \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.45, size = 69, normalized size = 1.9 \begin{align*}{\frac{-2\,i \left ( i\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) \right ) }{ade}\sqrt{e\cos \left ( dx+c \right ) }\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 [B] time = 2.30743, size = 103, normalized size = 2.86 \begin{align*} \frac{2 i \, \sqrt{-\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}}{\sqrt{a} d \sqrt{e} \sqrt{-\frac{2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04861, size = 165, normalized size = 4.58 \begin{align*} \frac{2 i \, \sqrt{2} \sqrt{\frac{1}{2}} \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{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{a d e} \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}{\sqrt{e \cos \left (d x + c\right )} \sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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