Optimal. Leaf size=31 \[ \frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.0230007, antiderivative size = 31, 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} \[ \frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \tan ^2(e+f x)}} \, dx &=\frac{\tan (e+f x) \int \cot (e+f x) \, dx}{\sqrt{b \tan ^2(e+f x)}}\\ &=\frac{\log (\sin (e+f x)) \tan (e+f x)}{f \sqrt{b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0816531, size = 39, normalized size = 1.26 \[ \frac{\tan (e+f x) (\log (\tan (e+f x))+\log (\cos (e+f x)))}{f \sqrt{b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 47, normalized size = 1.5 \begin{align*}{\frac{\tan \left ( fx+e \right ) \left ( 2\,\ln \left ( \tan \left ( fx+e \right ) \right ) -\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \right ) }{2\,f}{\frac{1}{\sqrt{b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63721, size = 45, normalized size = 1.45 \begin{align*} -\frac{\frac{\log \left (\tan \left (f x + e\right )^{2} + 1\right )}{\sqrt{b}} - \frac{2 \, \log \left (\tan \left (f x + e\right )\right )}{\sqrt{b}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85147, size = 119, normalized size = 3.84 \begin{align*} \frac{\sqrt{b \tan \left (f x + e\right )^{2}} \log \left (\frac{\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, b f \tan \left (f x + e\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 \frac{1}{\sqrt{b \tan ^{2}{\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 = 1.32646, size = 113, normalized size = 3.65 \begin{align*} -\frac{\frac{2 \, \log \left (-\frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1\right )}{\sqrt{b} \mathrm{sgn}\left (\tan \left (f x + e\right )\right )} - \frac{\log \left (-\frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )}{\sqrt{b} \mathrm{sgn}\left (\tan \left (f x + e\right )\right )}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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