Optimal. Leaf size=66 \[ -\frac{\cot (c+d x)}{2 b d \sqrt{b \tan ^2(c+d x)}}-\frac{\tan (c+d x) \log (\sin (c+d x))}{b d \sqrt{b \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.0291696, antiderivative size = 66, 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} \[ -\frac{\cot (c+d x)}{2 b d \sqrt{b \tan ^2(c+d x)}}-\frac{\tan (c+d x) \log (\sin (c+d x))}{b d \sqrt{b \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\left (b \tan ^2(c+d x)\right )^{3/2}} \, dx &=\frac{\tan (c+d x) \int \cot ^3(c+d x) \, dx}{b \sqrt{b \tan ^2(c+d x)}}\\ &=-\frac{\cot (c+d x)}{2 b d \sqrt{b \tan ^2(c+d x)}}-\frac{\tan (c+d x) \int \cot (c+d x) \, dx}{b \sqrt{b \tan ^2(c+d x)}}\\ &=-\frac{\cot (c+d x)}{2 b d \sqrt{b \tan ^2(c+d x)}}-\frac{\log (\sin (c+d x)) \tan (c+d x)}{b d \sqrt{b \tan ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.361502, size = 56, normalized size = 0.85 \[ -\frac{\tan ^3(c+d x) \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d \left (b \tan ^2(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 63, normalized size = 1. \begin{align*}{\frac{\tan \left ( dx+c \right ) \left ( \ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) \left ( \tan \left ( dx+c \right ) \right ) ^{2}-2\,\ln \left ( \tan \left ( dx+c \right ) \right ) \left ( \tan \left ( dx+c \right ) \right ) ^{2}-1 \right ) }{2\,d} \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 = 1.58767, size = 62, normalized size = 0.94 \begin{align*} \frac{\frac{\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{b^{\frac{3}{2}}} - \frac{2 \, \log \left (\tan \left (d x + c\right )\right )}{b^{\frac{3}{2}}} - \frac{1}{b^{\frac{3}{2}} \tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37851, size = 177, normalized size = 2.68 \begin{align*} -\frac{\sqrt{b \tan \left (d x + c\right )^{2}}{\left (\log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + \tan \left (d x + c\right )^{2} + 1\right )}}{2 \, b^{2} d \tan \left (d x + c\right )^{3}} \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 \frac{1}{\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 [B] time = 2.57734, size = 281, normalized size = 4.26 \begin{align*} -\frac{\frac{\mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{8 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} + \frac{4 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{4 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, b^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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