Optimal. Leaf size=50 \[ \frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)} \]
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Rubi [A] time = 0.0207209, antiderivative size = 50, 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} \[ \frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \sqrt{b \tan ^4(e+f x)} \, dx &=\left (\cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}\right ) \int \tan ^2(e+f x) \, dx\\ &=\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-\left (\cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}\right ) \int 1 \, dx\\ &=\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}\\ \end{align*}
Mathematica [A] time = 0.0914212, size = 41, normalized size = 0.82 \[ -\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)} \left (\tan ^{-1}(\tan (e+f x)) \cot (e+f x)-1\right )}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 42, normalized size = 0.8 \begin{align*} -{\frac{-\tan \left ( fx+e \right ) +\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f \left ( \tan \left ( fx+e \right ) \right ) ^{2}}\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 = 1.63842, size = 35, normalized size = 0.7 \begin{align*} -\frac{{\left (f x + e\right )} \sqrt{b} - \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 = 2.07455, size = 88, normalized size = 1.76 \begin{align*} -\frac{\sqrt{b \tan \left (f x + e\right )^{4}}{\left (f x - \tan \left (f x + e\right )\right )}}{f \tan \left (f x + e\right )^{2}} \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 \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 [B] time = 1.40617, size = 338, normalized size = 6.76 \begin{align*} \frac{{\left (\pi - 4 \, f x \tan \left (f x\right ) \tan \left (e\right ) - \pi \mathrm{sgn}\left (2 \, \tan \left (f x\right )^{2} \tan \left (e\right ) + 2 \, \tan \left (f x\right ) \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right ) - 2 \, \tan \left (e\right )\right ) \tan \left (f x\right ) \tan \left (e\right ) - \pi \tan \left (f x\right ) \tan \left (e\right ) + 2 \, \arctan \left (\frac{\tan \left (f x\right ) \tan \left (e\right ) - 1}{\tan \left (f x\right ) + \tan \left (e\right )}\right ) \tan \left (f x\right ) \tan \left (e\right ) + 2 \, \arctan \left (\frac{\tan \left (f x\right ) + \tan \left (e\right )}{\tan \left (f x\right ) \tan \left (e\right ) - 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) + 4 \, f x + \pi \mathrm{sgn}\left (2 \, \tan \left (f x\right )^{2} \tan \left (e\right ) + 2 \, \tan \left (f x\right ) \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right ) - 2 \, \tan \left (e\right )\right ) - 2 \, \arctan \left (\frac{\tan \left (f x\right ) \tan \left (e\right ) - 1}{\tan \left (f x\right ) + \tan \left (e\right )}\right ) - 2 \, \arctan \left (\frac{\tan \left (f x\right ) + \tan \left (e\right )}{\tan \left (f x\right ) \tan \left (e\right ) - 1}\right ) - 4 \, \tan \left (f x\right ) - 4 \, \tan \left (e\right )\right )} \sqrt{b}}{4 \,{\left (f \tan \left (f x\right ) \tan \left (e\right ) - f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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