Optimal. Leaf size=32 \[ -\frac{\cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0166386, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3658, 3475} \[ -\frac{\cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{b \tan ^2(c+d x)} \, dx &=\left (\cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=-\frac{\cot (c+d x) \log (\cos (c+d x)) \sqrt{b \tan ^2(c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.0389083, size = 32, normalized size = 1. \[ -\frac{\cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 37, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d\tan \left ( dx+c \right ) }\sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57723, size = 26, normalized size = 0.81 \begin{align*} \frac{\sqrt{b} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32182, size = 100, normalized size = 3.12 \begin{align*} -\frac{\sqrt{b \tan \left (d x + c\right )^{2}} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d \tan \left (d x + c\right )} \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 ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44285, size = 35, normalized size = 1.09 \begin{align*} \frac{\sqrt{b} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) \mathrm{sgn}\left (\tan \left (d x + c\right )\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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