Optimal. Leaf size=32 \[ -\frac {\cot (e+f x) \sqrt {b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, 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} \[ -\frac {\cot (e+f x) \sqrt {b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {b \tan ^2(e+f x)} \, dx &=\left (\cot (e+f x) \sqrt {b \tan ^2(e+f x)}\right ) \int \tan (e+f x) \, dx\\ &=-\frac {\cot (e+f x) \log (\cos (e+f x)) \sqrt {b \tan ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 1.00 \[ -\frac {\cot (e+f x) \sqrt {b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 38, normalized size = 1.19 \[ -\frac {\sqrt {b \tan \left (f x + e\right )^{2}} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 37, normalized size = 1.16 \[ \frac {\sqrt {b \left (\tan ^{2}\left (f x +e \right )\right )}\, \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \tan \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 19, normalized size = 0.59 \[ \frac {\sqrt {b} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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