Optimal. Leaf size=97 \[ -\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{b^2 f \sqrt{b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.0390056, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ -\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{b^2 f \sqrt{b \tan ^2(e+f x)}} \]
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 )^{5/2}} \, dx &=\frac{\tan (e+f x) \int \cot ^5(e+f x) \, dx}{b^2 \sqrt{b \tan ^2(e+f x)}}\\ &=-\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \int \cot ^3(e+f x) \, dx}{b^2 \sqrt{b \tan ^2(e+f x)}}\\ &=\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\tan (e+f x) \int \cot (e+f x) \, dx}{b^2 \sqrt{b \tan ^2(e+f x)}}\\ &=\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\log (\sin (e+f x)) \tan (e+f x)}{b^2 f \sqrt{b \tan ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.246315, size = 68, normalized size = 0.7 \[ \frac{\tan ^5(e+f x) \left (-\cot ^4(e+f x)+2 \cot ^2(e+f x)+4 \log (\tan (e+f x))+4 \log (\cos (e+f x))\right )}{4 f \left (b \tan ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 74, normalized size = 0.8 \begin{align*}{\frac{\tan \left ( fx+e \right ) \left ( 4\,\ln \left ( \tan \left ( fx+e \right ) \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{4}-2\,\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{4}+2\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}-1 \right ) }{4\,f} \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68978, size = 89, normalized size = 0.92 \begin{align*} -\frac{\frac{2 \, \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{b^{\frac{5}{2}}} - \frac{4 \, \log \left (\tan \left (f x + e\right )\right )}{b^{\frac{5}{2}}} - \frac{2 \, \sqrt{b} \tan \left (f x + e\right )^{2} - \sqrt{b}}{b^{3} \tan \left (f x + e\right )^{4}}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05207, size = 207, normalized size = 2.13 \begin{align*} \frac{{\left (2 \, \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 )^{4} + 3 \, \tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} - 1\right )} \sqrt{b \tan \left (f x + e\right )^{2}}}{4 \, b^{3} f \tan \left (f x + e\right )^{5}} \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{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.99926, size = 392, normalized size = 4.04 \begin{align*} -\frac{\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 )^{4} - 12 \, \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} + \frac{64 \, \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{32 \, \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{48 \, \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 )^{4} - 12 \, \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 )^{4}}}{64 \, b^{\frac{5}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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