Optimal. Leaf size=80 \[ \frac{2 i \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{4 i \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.208784, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3515, 3502, 3488} \[ \frac{2 i \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{4 i \sqrt{a+i a \tan (c+d x)} \sqrt{e \cos (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\left (\sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{2 i \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}+\frac{\left (2 \sqrt{e \cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{3 a}\\ &=\frac{2 i \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}}-\frac{4 i \sqrt{e \cos (c+d x)} \sqrt{a+i a \tan (c+d x)}}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.18951, size = 48, normalized size = 0.6 \[ \frac{2 (2 \tan (c+d x)-i) \sqrt{e \cos (c+d x)}}{3 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.364, size = 74, normalized size = 0.9 \begin{align*}{\frac{2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -4\,i}{3\,ad}\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 [A] time = 3.144, size = 108, normalized size = 1.35 \begin{align*} \frac{\sqrt{e}{\left (i \, \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) - 3 i \, \cos \left (\frac{1}{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 ) + \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sin \left (\frac{1}{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 )}}{3 \, \sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09765, size = 204, normalized size = 2.55 \begin{align*} \frac{\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}}{\left (-3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac{3}{2} i \, d x - \frac{3}{2} i \, c\right )}}{3 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e \cos{\left (c + d x \right )}}}{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )}}\, dx \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{\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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