Optimal. Leaf size=32 \[ \frac{2 \sqrt{b \tan (e+f x)}}{b f \sqrt{d \sec (e+f x)}} \]
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Rubi [A] time = 0.045485, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2605} \[ \frac{2 \sqrt{b \tan (e+f x)}}{b f \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2605
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}} \, dx &=\frac{2 \sqrt{b \tan (e+f x)}}{b f \sqrt{d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.38786, size = 32, normalized size = 1. \[ \frac{2 \sqrt{b \tan (e+f x)}}{b f \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 50, normalized size = 1.6 \begin{align*} 2\,{\frac{\sin \left ( fx+e \right ) }{f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{{\frac{d}{\cos \left ( fx+e \right ) }}}}}{\frac{1}{\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67059, size = 107, normalized size = 3.34 \begin{align*} \frac{2 \, \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt{\frac{d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{b d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.735, size = 51, normalized size = 1.59 \begin{align*} \begin{cases} \frac{2 \sqrt{\tan{\left (e + f x \right )}}}{\sqrt{b} \sqrt{d} f \sqrt{\sec{\left (e + f x \right )}}} & \text{for}\: f \neq 0 \\\frac{x}{\sqrt{b \tan{\left (e \right )}} \sqrt{d \sec{\left (e \right )}}} & \text{otherwise} \end{cases} \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{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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