Optimal. Leaf size=119 \[ -\frac {\tan (e+f x)}{b f \sqrt {b \tan ^4(e+f x)}}-\frac {x \tan ^2(e+f x)}{b \sqrt {b \tan ^4(e+f x)}}-\frac {\cot ^3(e+f x)}{5 b f \sqrt {b \tan ^4(e+f x)}}+\frac {\cot (e+f x)}{3 b f \sqrt {b \tan ^4(e+f x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ -\frac {x \tan ^2(e+f x)}{b \sqrt {b \tan ^4(e+f x)}}-\frac {\tan (e+f x)}{b f \sqrt {b \tan ^4(e+f x)}}-\frac {\cot ^3(e+f x)}{5 b f \sqrt {b \tan ^4(e+f x)}}+\frac {\cot (e+f x)}{3 b f \sqrt {b \tan ^4(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (b \tan ^4(e+f x)\right )^{3/2}} \, dx &=\frac {\tan ^2(e+f x) \int \cot ^6(e+f x) \, dx}{b \sqrt {b \tan ^4(e+f x)}}\\ &=-\frac {\cot ^3(e+f x)}{5 b f \sqrt {b \tan ^4(e+f x)}}-\frac {\tan ^2(e+f x) \int \cot ^4(e+f x) \, dx}{b \sqrt {b \tan ^4(e+f x)}}\\ &=\frac {\cot (e+f x)}{3 b f \sqrt {b \tan ^4(e+f x)}}-\frac {\cot ^3(e+f x)}{5 b f \sqrt {b \tan ^4(e+f x)}}+\frac {\tan ^2(e+f x) \int \cot ^2(e+f x) \, dx}{b \sqrt {b \tan ^4(e+f x)}}\\ &=\frac {\cot (e+f x)}{3 b f \sqrt {b \tan ^4(e+f x)}}-\frac {\cot ^3(e+f x)}{5 b f \sqrt {b \tan ^4(e+f x)}}-\frac {\tan (e+f x)}{b f \sqrt {b \tan ^4(e+f x)}}-\frac {\tan ^2(e+f x) \int 1 \, dx}{b \sqrt {b \tan ^4(e+f x)}}\\ &=\frac {\cot (e+f x)}{3 b f \sqrt {b \tan ^4(e+f x)}}-\frac {\cot ^3(e+f x)}{5 b f \sqrt {b \tan ^4(e+f x)}}-\frac {\tan (e+f x)}{b f \sqrt {b \tan ^4(e+f x)}}-\frac {x \tan ^2(e+f x)}{b \sqrt {b \tan ^4(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 45, normalized size = 0.38 \[ -\frac {\tan (e+f x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(e+f x)\right )}{5 f \left (b \tan ^4(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 62, normalized size = 0.52 \[ -\frac {{\left (15 \, f x \tan \left (f x + e\right )^{5} + 15 \, \tan \left (f x + e\right )^{4} - 5 \, \tan \left (f x + e\right )^{2} + 3\right )} \sqrt {b \tan \left (f x + e\right )^{4}}}{15 \, b^{2} f \tan \left (f x + e\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.25, size = 131, normalized size = 1.10 \[ -\frac {\frac {480 \, {\left (f x + e\right )}}{\sqrt {b}} - \frac {3 \, b^{\frac {9}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, b^{\frac {9}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 330 \, b^{\frac {9}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{b^{5}} + \frac {330 \, \sqrt {b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 35 \, \sqrt {b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, \sqrt {b}}{b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}}{480 \, b f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 63, normalized size = 0.53 \[ -\frac {\tan \left (f x +e \right ) \left (15 \arctan \left (\tan \left (f x +e \right )\right ) \left (\tan ^{5}\left (f x +e \right )\right )+15 \left (\tan ^{4}\left (f x +e \right )\right )-5 \left (\tan ^{2}\left (f x +e \right )\right )+3\right )}{15 f \left (b \left (\tan ^{4}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 50, normalized size = 0.42 \[ -\frac {\frac {15 \, {\left (f x + e\right )}}{b^{\frac {3}{2}}} + \frac {15 \, \tan \left (f x + e\right )^{4} - 5 \, \tan \left (f x + e\right )^{2} + 3}{b^{\frac {3}{2}} \tan \left (f x + e\right )^{5}}}{15 \, f} \]
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 \[ \int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \tan ^{4}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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