Optimal. Leaf size=41 \[ \frac{i \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}} \]
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Rubi [A] time = 0.0955962, antiderivative size = 41, 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, 37} \[ \frac{i \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.13828, size = 41, normalized size = 1. \[ \frac{i \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 65, normalized size = 1.6 \begin{align*}{\frac{-i \left ( 1+i\tan \left ( fx+e \right ) \right ) }{fa \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 = 1.72384, size = 47, normalized size = 1.15 \begin{align*} \frac{i \, \sqrt{c} \sqrt{-i \, \tan \left (f x + e\right ) + 1}}{\sqrt{a} f \sqrt{i \, \tan \left (f x + e\right ) + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34277, size = 223, normalized size = 5.44 \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 (3 i \, f x + 3 i \, e\right )} + i \, e^{\left (2 i \, f x + 2 i \, e\right )} - i \, e^{\left (i \, f x + i \, e\right )} + i\right )} e^{\left (-i \, f x - i \, e\right )}}{a 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{\sqrt{- c \left (i \tan{\left (e + f x \right )} - 1\right )}}{\sqrt{a \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{\sqrt{-i \, c \tan \left (f x + e\right ) + c}}{\sqrt{i \, a \tan \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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