Optimal. Leaf size=44 \[ \frac{\tan (e+f x)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.0998342, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {3523, 39} \[ \frac{\tan (e+f x)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.26144, size = 64, normalized size = 1.45 \[ \frac{\sin (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x))}{c f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.089, size = 82, normalized size = 1.9 \begin{align*}{\frac{ \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \tan \left ( fx+e \right ) }{fac \left ( \tan \left ( fx+e \right ) +i \right ) ^{2} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{2}}\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80062, size = 22, normalized size = 0.5 \begin{align*} \frac{\sin \left (f x + e\right )}{\sqrt{a} \sqrt{c} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29461, size = 171, normalized size = 3.89 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-i \, e^{\left (4 i \, f x + 4 i \, e\right )} + i\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a c f} \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{1}{\sqrt{a \left (i \tan{\left (e + f x \right )} + 1\right )} \sqrt{- c \left (i \tan{\left (e + f 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{1}{\sqrt{i \, a \tan \left (f x + e\right ) + a} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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