Optimal. Leaf size=50 \[ \frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)} \]
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Rubi [A] time = 0.020658, 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} \[ \frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \sqrt{b \tan ^4(c+d x)} \, dx &=\left (\cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-\left (\cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)}}{d}-x \cot ^2(c+d x) \sqrt{b \tan ^4(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.0884631, size = 41, normalized size = 0.82 \[ -\frac{\cot (c+d x) \sqrt{b \tan ^4(c+d x)} \left (\tan ^{-1}(\tan (c+d x)) \cot (c+d x)-1\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 42, normalized size = 0.8 \begin{align*} -{\frac{-\tan \left ( dx+c \right ) +\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d \left ( \tan \left ( dx+c \right ) \right ) ^{2}}\sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49983, size = 35, normalized size = 0.7 \begin{align*} -\frac{{\left (d x + c\right )} \sqrt{b} - \sqrt{b} \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34459, size = 88, normalized size = 1.76 \begin{align*} -\frac{\sqrt{b \tan \left (d x + c\right )^{4}}{\left (d x - \tan \left (d x + c\right )\right )}}{d \tan \left (d x + c\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 (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.42438, size = 309, normalized size = 6.18 \begin{align*} \frac{{\left (\pi - 4 \, d x \tan \left (d x\right ) \tan \left (c\right ) - \pi \mathrm{sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) \tan \left (d x\right ) \tan \left (c\right ) - \pi \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac{\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac{\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 4 \, d x + \pi \mathrm{sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) - 2 \, \arctan \left (\frac{\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) - 2 \, \arctan \left (\frac{\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) - 4 \, \tan \left (d x\right ) - 4 \, \tan \left (c\right )\right )} \sqrt{b}}{4 \,{\left (d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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