Optimal. Leaf size=61 \[ \frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.027738, antiderivative size = 61, normalized size of antiderivative = 1., 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} \[ \frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}+\frac{b \cot (c+d x) \sqrt{b \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (b \tan ^2(c+d x)\right )^{3/2} \, dx &=\left (b \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan ^3(c+d x) \, dx\\ &=\frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}-\left (b \cot (c+d x) \sqrt{b \tan ^2(c+d x)}\right ) \int \tan (c+d x) \, dx\\ &=\frac{b \cot (c+d x) \log (\cos (c+d x)) \sqrt{b \tan ^2(c+d x)}}{d}+\frac{b \tan (c+d x) \sqrt{b \tan ^2(c+d x)}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.115158, size = 47, normalized size = 0.77 \[ \frac{\cot ^3(c+d x) \left (b \tan ^2(c+d x)\right )^{3/2} \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 48, normalized size = 0.8 \begin{align*} -{\frac{- \left ( \tan \left ( dx+c \right ) \right ) ^{2}+\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d \left ( \tan \left ( dx+c \right ) \right ) ^{3}} \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49755, size = 46, normalized size = 0.75 \begin{align*} \frac{b^{\frac{3}{2}} \tan \left (d x + c\right )^{2} - b^{\frac{3}{2}} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37052, size = 135, normalized size = 2.21 \begin{align*} \frac{{\left (b \tan \left (d x + c\right )^{2} + b \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + b\right )} \sqrt{b \tan \left (d x + c\right )^{2}}}{2 \, d \tan \left (d x + c\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 (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.51155, size = 55, normalized size = 0.9 \begin{align*} \frac{1}{2} \, b^{\frac{3}{2}}{\left (\frac{\tan \left (d x + c\right )^{2}}{d} - \frac{\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d}\right )} \mathrm{sgn}\left (\tan \left (d x + c\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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