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 = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, 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 3473
Rule 3475
Rule 3658
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*}
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Mathematica [A] time = 0.26, size = 68, normalized size = 0.70 \[ \frac {\tan ^5(e+f x) \left (-\cot ^4(e+f x)+2 \cot ^2(e+f x)+4 \log (\tan (e+f x))+4 \log (\cos (e+f x))\right )}{4 f \left (b \tan ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 82, normalized size = 0.85 \[ \frac {{\left (2 \, \log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} - 1\right )} \sqrt {b \tan \left (f x + e\right )^{2}}}{4 \, b^{3} f \tan \left (f x + e\right )^{5}} \]
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 = 0.35, size = 74, normalized size = 0.76 \[ \frac {\tan \left (f x +e \right ) \left (4 \ln \left (\tan \left (f x +e \right )\right ) \left (\tan ^{4}\left (f x +e \right )\right )-2 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (\tan ^{4}\left (f x +e \right )\right )+2 \left (\tan ^{2}\left (f x +e \right )\right )-1\right )}{4 f \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 66, normalized size = 0.68 \[ -\frac {\frac {2 \, \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{b^{\frac {5}{2}}} - \frac {4 \, \log \left (\tan \left (f x + e\right )\right )}{b^{\frac {5}{2}}} - \frac {2 \, \sqrt {b} \tan \left (f x + e\right )^{2} - \sqrt {b}}{b^{3} \tan \left (f x + e\right )^{4}}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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