Optimal. Leaf size=98 \[ \frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}-\frac{b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
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Rubi [A] time = 0.0390226, antiderivative size = 98, 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} \[ \frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}-\frac{b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (b \tan ^2(e+f x)\right )^{5/2} \, dx &=\left (b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)}\right ) \int \tan ^5(e+f x) \, dx\\ &=\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}-\left (b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)}\right ) \int \tan ^3(e+f x) \, dx\\ &=-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}+\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}+\left (b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)}\right ) \int \tan (e+f x) \, dx\\ &=-\frac{b^2 \cot (e+f x) \log (\cos (e+f x)) \sqrt{b \tan ^2(e+f x)}}{f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}+\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}\\ \end{align*}
Mathematica [A] time = 0.372675, size = 56, normalized size = 0.57 \[ -\frac{\cot (e+f x) \left (b \tan ^2(e+f x)\right )^{5/2} \left (2 \cot ^2(e+f x)+4 \cot ^4(e+f x) \log (\cos (e+f x))-1\right )}{4 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.146, size = 58, normalized size = 0.6 \begin{align*}{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{4}-2\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}+2\,\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }{4\,f \left ( \tan \left ( fx+e \right ) \right ) ^{5}} \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 = 1.6261, size = 63, normalized size = 0.64 \begin{align*} \frac{b^{\frac{5}{2}} \tan \left (f x + e\right )^{4} - 2 \, b^{\frac{5}{2}} \tan \left (f x + e\right )^{2} + 2 \, b^{\frac{5}{2}} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93873, size = 180, normalized size = 1.84 \begin{align*} \frac{{\left (b^{2} \tan \left (f x + e\right )^{4} - 2 \, b^{2} \tan \left (f x + e\right )^{2} - 2 \, b^{2} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, b^{2}\right )} \sqrt{b \tan \left (f x + e\right )^{2}}}{4 \, f \tan \left (f x + e\right )} \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 \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 [A] time = 1.4196, size = 80, normalized size = 0.82 \begin{align*} \frac{1}{4} \, b^{\frac{5}{2}}{\left (\frac{2 \, \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{f} + \frac{f \tan \left (f x + e\right )^{4} - 2 \, f \tan \left (f x + e\right )^{2}}{f^{2}}\right )} \mathrm{sgn}\left (\tan \left (f x + e\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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