Optimal. Leaf size=51 \[ -\frac{x \tan ^2(e+f x)}{\sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}} \]
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Rubi [A] time = 0.0211575, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ -\frac{x \tan ^2(e+f x)}{\sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \tan ^4(e+f x)}} \, dx &=\frac{\tan ^2(e+f x) \int \cot ^2(e+f x) \, dx}{\sqrt{b \tan ^4(e+f x)}}\\ &=-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}}-\frac{\tan ^2(e+f x) \int 1 \, dx}{\sqrt{b \tan ^4(e+f x)}}\\ &=-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}}-\frac{x \tan ^2(e+f x)}{\sqrt{b \tan ^4(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0503223, size = 43, normalized size = 0.84 \[ -\frac{\tan (e+f x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(e+f x)\right )}{f \sqrt{b \tan ^4(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 40, normalized size = 0.8 \begin{align*} -{\frac{\tan \left ( fx+e \right ) \left ( \arctan \left ( \tan \left ( fx+e \right ) \right ) \tan \left ( fx+e \right ) +1 \right ) }{f}{\frac{1}{\sqrt{b \left ( \tan \left ( fx+e \right ) \right ) ^{4}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53725, size = 36, normalized size = 0.71 \begin{align*} -\frac{\frac{f x + e}{\sqrt{b}} + \frac{1}{\sqrt{b} \tan \left (f x + e\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01841, size = 93, normalized size = 1.82 \begin{align*} -\frac{\sqrt{b \tan \left (f x + e\right )^{4}}{\left (f x \tan \left (f x + e\right ) + 1\right )}}{b f \tan \left (f x + e\right )^{3}} \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{b \tan ^{4}{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53148, size = 65, normalized size = 1.27 \begin{align*} -\frac{\frac{2 \,{\left (f x + e\right )}}{\sqrt{b}} - \frac{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\sqrt{b}} + \frac{1}{\sqrt{b} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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