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 = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 3473
Rule 3475
Rule 3658
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*}
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Mathematica [A] time = 0.36, 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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fricas [A] time = 0.41, size = 69, normalized size = 1.05 \[ -\frac {\sqrt {b \tan \left (f x + e\right )^{2}} {\left (\log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{2} + \tan \left (f x + e\right )^{2} + 1\right )}}{2 \, b^{2} f \tan \left (f x + e\right )^{3}} \]
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 = 0.31, size = 64, normalized size = 0.97 \[ -\frac {\tan \left (f x +e \right ) \left (2 \ln \left (\tan \left (f x +e \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )-\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )+1\right )}{2 f \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 46, normalized size = 0.70 \[ \frac {\frac {\log \left (\tan \left (f x + e\right )^{2} + 1\right )}{b^{\frac {3}{2}}} - \frac {2 \, \log \left (\tan \left (f x + e\right )\right )}{b^{\frac {3}{2}}} - \frac {1}{b^{\frac {3}{2}} \tan \left (f x + e\right )^{2}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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