Optimal. Leaf size=66 \[ -\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \log (\sin (e+f x))}{b f \sqrt{b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.0368908, antiderivative size = 66, 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, 3475} \[ -\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \log (\sin (e+f x))}{b f \sqrt{b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\left (b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\tan (e+f x) \int \cot ^3(e+f x) \, dx}{b \sqrt{b \tan ^2(e+f x)}}\\ &=-\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \int \cot (e+f x) \, dx}{b \sqrt{b \tan ^2(e+f x)}}\\ &=-\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\log (\sin (e+f x)) \tan (e+f x)}{b f \sqrt{b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.391674, size = 56, normalized size = 0.85 \[ -\frac{\tan ^3(e+f x) \left (\cot ^2(e+f x)+2 \log (\tan (e+f x))+2 \log (\cos (e+f x))\right )}{2 f \left (b \tan ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 64, normalized size = 1. \begin{align*} -{\frac{\tan \left ( fx+e \right ) \left ( 2\,\ln \left ( \tan \left ( fx+e \right ) \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{2}-\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{2}+1 \right ) }{2\,f} \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64679, size = 62, normalized size = 0.94 \begin{align*} \frac{\frac{\log \left (\tan \left (f x + e\right )^{2} + 1\right )}{b^{\frac{3}{2}}} - \frac{2 \, \log \left (\tan \left (f x + e\right )\right )}{b^{\frac{3}{2}}} - \frac{1}{b^{\frac{3}{2}} \tan \left (f x + e\right )^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89531, size = 177, normalized size = 2.68 \begin{align*} -\frac{\sqrt{b \tan \left (f x + e\right )^{2}}{\left (\log \left (\frac{\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} + \tan \left (f x + e\right )^{2} + 1\right )}}{2 \, b^{2} f \tan \left (f x + e\right )^{3}} \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}{\left (b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.62848, size = 300, normalized size = 4.55 \begin{align*} -\frac{\frac{\mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} - \frac{8 \, \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} + \frac{4 \, \log \left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} - \frac{4 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}}{8 \, b^{\frac{3}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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